List of Mini-Symposia

This minisymposium aims to bring together leading researchers working on the analysis and application of hybrid and polytopal discretization methods. Particular emphasis will be placed on the development of novel numerical schemes for coupled partial differential equations (PDEs) arising in complex multiphysics problems, including interface-coupled systems and heterogeneous media. The session will cover recent advances in the design, analysis, and implementation of discretization techniques for problems in continumm mechanics. Contributions addressing spectral approximations and eigenvalue problems associated with these formulations are also strongly encouraged.

Complex multiphysics flows involving free surfaces, fluid–structure interaction, granular materials, fracture, and multi-phase coupling arise in many engineering and geophysical applications, including coastal and hydraulic engineering, dam-break flows, sediment transport, debris flows, and impact problems. Accurate and robust numerical simulation of these problems remains highly challenging due to strong nonlinearities, moving interfaces, large deformations, and multi-scale interactions. In recent years, high-order numerical methods and particle-based approaches have demonstrated significant potential for addressing these challenges. High-order discretization techniques provide improved accuracy and numerical efficiency for complex flow structures, while particle methods offer natural advantages in handling large deformations, free surfaces, fracture, and evolving interfaces. Increasing attention has therefore been devoted to the development of advanced computational frameworks that combine high-order accuracy, numerical stability, conservation properties, and multiphysics coupling capabilities. This minisymposium aims to bring together researchers working on high-order and particle-based numerical methods for complex multiphysics flows. Topics of interest include, but are not limited to: Smoothed Particle Hydrodynamics (SPH), meshfree and hybrid numerical methods, fluid–structure interaction, multi-phase and granular flows, free-surface and dam-break problems, fracture and large deformation modeling, geomechanics and hydraulic engineering applications, and high-performance computing for multiphysics simulations. The minisymposium will provide a platform for discussing recent theoretical developments, numerical algorithms, and engineering applications, while promoting interdisciplinary interactions between researchers in computational fluid dynamics, computational mechanics, applied mathematics, and scientific computing.

High-order accurate and nonlinearly stable numerical methods are especially efficient for hyperbolic or convection-dominated partial differential equation (PDE) models, which contain both singularities and complex smooth solution structures. This minisymposium focuses on recent developments and applications of high-order numerical methods, especially weighted essentially non-oscillatory (WENO) finite difference and finite volume methods, and discontinuous Galerkin (DG) finite element methods, for hyperbolic and convection-dominated problems. Topics include algorithmic efficiency for high-order accurate methods and high-dimensional problems; theoretical analysis of high-order schemes; positivity-preserving, well-balanced, and other structure-preserving approaches; novel time-discretization methods; effective methods for ensuring the nonlinear stability of high-order schemes; etc. This minisymposium is also dedicated to Professor Chi-Wang Shu on the occasion of his 70th birthday, recognizing the profound impact of his work and celebrating his achievements.

In numerical approximation of PDEs, the design of a numerical method is justified by a priori error estimates, while its actual behavior is controlled by (and possibly steered by) a posteriori error estimates. Besides, to practically obtain the numerical approximation, a suitable solver for the associated system of nonlinear or linear algebraic equations is needed. Numerical parameters (such as mesh size and regularity, polynomial degree, or amount of stabilization) and PDE parameters (such as mutual ratios of advective, reactive, and diffusive effects, wavenumber, strength of the nonlinearity, or final time size for evolution problems) enter the numerical analysis and can affect the actual performance of such methods. When the analysis and/or the practical performance of these methods is insensitive to these parameters, we speak of robustness. The subject of this minisymposium is robustness in a priori and a posteriori error estimates and in (iterative) algebraic solvers.

High-order discretization methods — including high-order finite elements and Isogeometric Analysis (IGA) — have established themselves as powerful paradigms for the numerical solution of partial differential equations, offering superior accuracy per degree of freedom and, in the case of IGA, exact geometry representation through NURBS and related spline technologies. A hallmark of these approaches is the use of high polynomial degrees, which poses unique challenges for solver design but also opens new opportunities for high-performance computing. This minisymposium focuses on the design, analysis, and implementation of scalable preconditioners and solvers tailored to high-order finite element and isogeometric discretizations, with emphasis on robustness with respect to the polynomial degree p, scalability with respect to the mesh size h, and parallel efficiency on modern HPC architectures. A broad spectrum of algorithmic approaches is within scope, including Domain Decomposition Methods (DDM) such as FETI, BDDC, and Schwarz methods; multigrid and multilevel preconditioners; low-rank and hierarchical solvers; and tensor-product techniques that exploit the structure of spline discretizations. A key computational advantage of high-order discretizations is that increasing the polynomial degree yields denser local stiffness matrices compared to low-order methods on equivalent meshes, resulting in a more favorable computation-to-communication ratio. This makes high-order finite elements and IGA particularly well-suited for massively parallel execution, provided that solvers and preconditioners are designed to exploit this structure effectively. Contributions will span p-robust and h-robust preconditioners; spectral analysis of solver operators in the high-order and spline setting; multilevel and coarse space construction strategies; multi-patch and non-conforming coupling strategies; and scalability studies on large-scale parallel architectures. Both theoretical foundations and practical algorithmic developments are welcome. The minisymposium aims to bring together researchers from numerical linear algebra, computational mechanics, and high-performance computing to discuss recent advances and open problems in this rapidly evolving area.

Pointwise estimates and related techniques (W^{1,p} estimates, etc) are important in numerical analysis of partial differential equations. They are not only critical in pointwise tracking optimal control, but also applicable in several nonlinear PDEs (incompressible Magnetohydrodynamics, etc). In this minisymposium, speakers will focus on pointwise estimates and related techniques for hybridizable numerical methods like hybridizable discontinuous Galerkin method, hybrid high-order method, virtual element method, etc. We would like to emphasize that due to the non-conformity of these numerical methods, significant theoretic difficulties may rise, such that these estimates for hybridizable numerical methods are not simple analogues of those of conforming methods.

Boundary integral equations have nowadays reached a high level of consideration to model several problems in engineering and applied sciences. In parallel, various numerical methods have been introduced and analysed to offer accurate and efficient approaches for the approximate resolution of such type of equations. This minisymposium aims at presenting recent advances in this framework, ranging from high order boundary elements for scalar and vector wave equations to efficient BEMs in industrial applications.

Integral operators play a key role in a wide range of mathematical models, including boundary integral formulations and nonlocal problems such as fractional differential equations. Their discretization by means of the boundary element method (BEM) or finite element techniques pose significant challenges due to singular kernels, nonlocal interactions, dense matrices, and reduced (boundary) regularity of solutions. Recently, many contributions to counteract these difficulties have been made that enable the accurate simulation of integral equations. The goal of this minisymposium is to bring together researchers currently working on the numerical analysis and implementation of boundary integral and fractional differential operators, with a particular focus on high-order methods and efficient discretizations such as adaptive mesh refinement, quadrature techniques, kernel approximation, and operator compression.

The development of finite element discretizations for partial differential equations (PDEs) on non-body-fitted meshes has attracted significant attention due to their flexibility in handling complex and evolving geometries. In these approaches, the PDE is posed on a physical domain, while the numerical discretization is carried out on a computational mesh that does not conform to the boundaries or interfaces of the physical domain. Such methods are particularly well suited for problems involving evolving domains, interfaces, or topological changes. However, this geometric mismatch introduces several analytical and numerical challenges, especially in the design of stable and accurate high-order methods. This minisymposium aims to bring together researchers working on unfitted and related discretization techniques, including fictitious domain methods, CutFEM, X-FEM, $\phi$-FEM, the Shifted Boundary Method, the Transfer Path Method, HHO methods, and the Immersed Boundary Method, among others. Particular emphasis will be placed on recent advances and open challenges in high-order unfitted discretizations, as well as on the analysis, implementation, and applications of such methods more broadly.

The aim of this minisymposium is to discuss recent progress in the analysis and application of high-order methods for approximating the parameter-to-solution map in the context of random and parametric PDEs; such problems arise e.g. in uncertainty quantification and PDE-based optimization and control. We welcome theoretical, methodological, and application-oriented contributions on, among others, polynomial/rational approximations, kernel and Gaussian process regression methods, low-rank, proper generalized decomposition and reduced order models, as well as scientific-machine-learning-based methods for operator learning.

Boundary integral equations (BIEs) are a useful framework for reducing partial differential equations to the boundaries and interfaces of a physical domain. Though these methods result in dense matrices and typically require an explicit Green's function, recent advances in the analysis and implementation of these methods have made it easier to handle complex wave problems and incorporate new physics such as hydro-, elasto-, and electrodynamics. This session highlights these advances by bringing together recent work on both mathematical formulations of BIEs and fast computational tools for solving them. Topics include skeletonization and hierarchical compression, broadband solvers, periodic summation techniques, Maxwell waveguides, and ice-ocean interactions.

Phase-field models are widely used to capture interface dynamics in new materials, lithium batteries and biomolecular systems, etc. This minisymposium focuses on state-of-the-art high-order structure-preserving algorithms that rigorously preserve intrinsic physical structures of phase-field models. All contributions include complete theoretical analysis and practical numerical implementations. We intend to connect theorists and application-oriented computational scientists to address open challenges in constructing high-order, linear-decoupled, adaptive structure-preserving schemes for industrial multi-scale simulations.

Many systems of partial differential equations arising in physics and engineering are endowed with intrinsic geometric, analytic, and thermodynamic structures. These include involution constraints, such as the divergence-free condition of magnetic fields or the curl-free character of deformation tensor, thermodynamic compatibility requirements encoded in energy conservation and entropy balance or inequalities, and physical admissibility constraints such as positivity of density and pressure. At the same time, these PDE systems are often genuinely multiscale. Important examples include low-Mach-number flows, stiff relaxation systems, and hyperbolic reformulations of parabolic models, where the correct limiting behavior must be recovered in asymptotic regimes. These features make the design of reliable numerical methods particularly demanding: schemes should be accurate and robust, while preserving, at the discrete level, the fundamental structural properties of the continuous problem. The aim of this minisymposium is to review recent advances in high order structure-preserving methods for multiscale PDE systems, with emphasis on the preservation of involutions, thermodynamic compatibility, asymptotic limits, and physical admissibility constraints. Contributions will cover both methodological developments and applications, including continuum mechanics, plasma flows, electromagnetics, and multi-material flows.

High-order polytopal methods have emerged as a powerful framework for the numerical approximation of partial differential equations (PDEs), combining high-order accuracy with the flexibility of general polygonal and polyhedral meshes. This mini-symposium, organized in the framework of the ERC SyG NEMESIS project, brings together researchers working on the analysis, development, and application of advanced, high-order, polytopal discretization techniques, including Virtual Elements, Hybrid High-Order, (hybridized) polytopal discontinuous Galerkin, Discrete De Rham methods, and related approaches. Contributions will address theoretical advances, efficient algorithms, adaptivity, machine-learning enhancement, and applications to challenging problems in science and engineering. The session aims to foster exchanges between experts in numerical analysis, scientific computing, and applied mathematics, highlighting recent developments and future directions in high-order polytopal methods.

The application of Extreme Learning Machines (ELMs) to the numerical solution of differential equations has experienced remarkable growth in recent years. Starting from the pioneering Physics-Informed Extreme Learning Machine (PIELM) framework, an increasingly broad family of methods has emerged, extending the original formulation to more complex differential models, inverse problems, domain decomposition strategies and adaptive techniques. At the same time, several approaches originally developed under different names and motivations can be interpreted within the same random projection paradigm, highlighting a common mathematical structure that deserves further investigation. This mini-symposium aims to bring together researchers working on ELM-based and related random projection methods for differential equations, with the objective of discussing their theoretical foundations, computational properties, and practical applications. Particular attention will be devoted to the convergence behavior of these methods. Although they are not high-order methods in the classical polynomial approximation sense, growing theoretical and numerical evidence indicates that they often exhibit spectral-like convergence, placing them among the closest physics-informed learning approaches to traditional spectral methods. This distinctive behavior originates from the use of a fixed, randomly generated hidden layer that defines a rich approximation space while preserving the computational efficiency characteristic of ELMs. The mini-symposium welcomes contributions addressing theoretical analysis, numerical algorithms, and applications of ELM-based methods for ordinary, partial, stochastic and integro-differential equations. Contributions employing ELMs or related random feature techniques as components of broader numerical frameworks are equally encouraged. The goal is to provide a forum where researchers can exchange ideas, identify common principles underlying existing methods, and discuss future directions for this rapidly evolving research area.

The accurate numerical approximation of partial differential equations on complex geometries remains a major challenge in scientific computing. General polygonal and polyhedral meshes, curved boundaries, heterogeneous materials, and multiphysics interfaces require discretization techniques that combine geometric flexibility, robustness, and high-order accuracy. This mini-symposium aims to highlight recent advances in high-order numerical methods, from the perspective of early-career researchers. Contributions may involve finite element, virtual element method, spectral element method, discontinuous Galerkin, hybrid-high order, isogeometric analysis, together with developments in adaptivity and refinement strategies. Particular attention will be devoted to the application of these approaches to challenging problems, including eigenvalue problems, multiscale and multiphysics models, contact and interface phenomena, and fluid–structure interaction. These applications raise fundamental questions related to accuracy, stability, and computational efficiency on complex domains. Particular emphasis will be placed on novel ideas and emerging trends, such as the interplay between standard methods and machine learning techniques, that exploit high-order discretizations to achieve accurate and efficient simulations on general computational meshes. By bringing together young scientists working across different numerical frameworks, the mini-symposium seeks to foster interactions among research communities and provide a snapshot of emerging trends and future directions on high-order computational methods for PDEs.

The aim of this minisymposium is to discuss the role of Hilbert complexes and related structure-preserving discretizations in the analysis, and implementation of high order and high performance numerical methods for low-frequency electromagnetic simulations and related PDEs. We would like to cover a wide range of approaches: from methods well stablished in computational electromagnetism, like Whitney forms and finite element exterior calculus, discontinuous Galerkin or finite volume methods, to more recent approaches like polytopal methods, virtual element methods or isogeometric methods. Applications to other PDEs with geometric or topological structure, as well as computational aspects connected with structure preservation, such as domain decomposition, high-performance implementations, and model-reduction techniques, will also be considered.

Wave scattering problems in acoustics, elastodynamics, and electromagnetics arise across a broad spectrum of applications, where intricate mathematical challenges and numerical complexities often demand sophisticated methodologies. Substantial research efforts have been devoted to numerical solution strategies by both mathematical and engineering communities. This symposium aims to bring together researchers to exchange ideas and discuss recent advances in the computational modeling and simulation of complex wave scattering problems utilizing high-order schemes.

Fourth-order partial differential equations arise in a wide range of applications, from thin plate and shell mechanics to phase-field models, fluid dynamics, and biological membranes. Their numerical approximation poses significant challenges, particularly when trying to avoid the construction of C1 conforming finite element spaces, which can be complex and restrictive. Nonstandard finite element methods - such as discontinuous Galerkin, C0 interior penalty, mixed formulations, virtual element methods, and Hybrid High Order methods - offer flexible alternatives that accommodate general meshes, local adaptivity, and robust handling of singularities and interfaces. In this minisymposium, we aim to discuss recent advances in the design, analysis, and efficient implementation of these nonstandard discretizations for both fourth-order problems and their extension to problems that require (local) H2-regularity. Topics include, but are not limited to, novel a priori and a posteriori error analysis, hp-adaptivity, structure-preserving schemes and the application of such methods to emerging problems in engineering and the sciences.

The Active Flux method is a hybrid finite element – finite volume method that achieves upwinding by locally evolving continuous data, instead of solving Riemann problems. Typically its degrees of freedom are cell averages and point values located at cell interfaces, the latter being shared between adjacent cells. The solution has a globally continuous representation. The evolution of the averages is conservative, which means this is a finite volume method that is able to converge to the weak solution. This method was proposed by Phil Roe around 15 years ago. He suggested a 3rd order method that can be found in the literature under the name of ‘Active Flux’. Since then it has been shown to achieve superior results on coarse grids, because it is preserves multi-dimensional structures of the flow. In recent developments Active Flux was made arbitrarily high-order accurate. In this mini-symposium we want to study theoretical properties of this method, its various numerical implementations and how the method fares in applications. Theoretical one has been able to prove structure preserving properties of the method for linearized equations, so how to proceed for non-linear equations? One has understood some relationship between different numerical implementations of the method, so which version is best in a given context? Finally the application of Active Flux to computational astrophysics has made progress with very promising results, this needs to be pushed further. This mini-symposium will bring together experts to discuss these topics and move the subject forward.

Many scientific and engineering applications involve phenomena evolving across multiple, possibly non-separated, spatial and temporal scales, posing significant challenges for numerical simulation. This minisymposium brings together recent advances in the design, analysis, and implementation of specialized numerical methods for such multiscale problems, highlighting complementary perspectives and methodologies. Topics include algorithms that accurately capture fine-scale effects while remaining computationally tractable. Particular emphasis is placed on high-order techniques; examples include higher-order multiscale approximation spaces, optimal $hp$-approximation, and high-order expansions in time. Applications range from the propagation of waves and the flow of fluids in heterogeneous media to mechanics and quantum physics, as well as network dynamics. These applications illustrate how modern numerical methods can be used to reliably simulate complex multiscale phenomena that would otherwise be too computationally intensive. The minisymposium aims to bring together researchers from different countries and research directions, fostering the exchange of ideas and establishing new collaborations on high-order numerical methods for taming multiple scales.

This minisymposium invites contributions on cutting-edge developments in the numerical approximation of time-dependent partial differential equations. The focus is on innovative high-order space-time discretization techniques, including finite element methods and isogeometric analysis, as well as domain decomposition and parallelization strategies, and advanced iterative solvers and preconditioners. The minisymposium aims to foster exchange between theoretical advances and real-world applications, bringing together researchers interested in pushing the boundaries of large-scale computational science.

Relativistic quantum mechanical models form an important class of dispersive partial differential equations, with applications ranging from quantum physics to materials science and high-field phenomena. Their numerical simulation presents substantial challenges, including highly oscillatory dynamics, multiscale regimes, spinor coupling, long-time accuracy, and the preservation of intrinsic mathematical structures. This mini-symposium brings together computational and applied mathematicians and computational physicists working on spectral and high-order numerical methods for relativistic quantum models. The focus will be on accurate and efficient discretizations, robust time integration strategies, and rigorous numerical analysis for Dirac-type, Klein-Gordon-type and relativistic Schroedinger-type as well as their related coupled systems. Particular emphasis will be placed on how structure-preserving spectral and high-order methods can improve resolution, stability, and efficiency across different physical regimes.

From classical rarefied gas dynamics to modern socio-economic and biological modeling, understanding non-equilibrium behavior relies heavily on capturing microscopic interactions of particles. Collisional kinetic equations represent the most appropriate models to describe such multiscale phenomena. However, because the overarching evolution of these systems is governed by complex collision integrals, accurately simulating these dynamics is extremely challenging. Thus, an accurate and fast treatment of these interactions is necessary for robust results. Several numerical methods have been developed to provide reliable approximations of collisional equations and deterministic methods have consistently shown strong robustness and stability. In this framework, spectral methods have emerged as a powerful tool. By exploiting Fourier representations, convolution-type formulations, and fast summation algorithms, spectral techniques enable the efficient evaluation of the collision operator while retaining high-order accuracy for smooth profiles and allowing the preservation of the main physical invariants. This minisymposium aims to present recent advances in various aspects of spectral methods for collisional kinetic equations, focusing on numerical properties and computational aspects, including the embedding of these methods in large-scale computations of rarefied gases and plasmas. We bring together researchers from different communities of the field, with the scope of encouraging discussions and providing insights for possible future collaborations.

The fast and accurate solution of wave propagation problems remains a challenging obstacle in many applications. Low-order polynomial methods typically deliver poor approximations due to dispersion errors, so that specific numerical schemes are required. This minisymposium will bring together a variety of researchers working on this topic, from the design and analysis of numerical methods to the implementation of solvers. A list of speakers who accepted to participate is given in the attached pdf file, together with tentative titles for their presentations.

High Order Mimetic difference methods are numerical techniques for solving partial differential equations by using discrete operators that mirror the essential properties of continuous mathematical and physical systems, such as conservation laws, sym- metry, and vector calculus identities. In addition, the Mimetic Operators Library Enhanced (MOLE) has been created to facilitate writing computationally efficient codes to solve complex societal problems. Here we present some recent advance- ment on the mimetic difference methods and some applications to demonstrate the effectiveness and applicability of these methods.

Integral equation methods have become increasingly practical and powerful for a wide range of problems with interfaces, including fluid dynamics, wave scattering, and electromagnetics. However, the dense linear systems that arise place a premium on accuracy per degree of freedom, making high-order methods and fast algorithms central components of efficient solvers. Recent developments, including high-order quadrature for near- and weakly-singular integrals, geometry-aware discretizations, fast direct solvers, and fast summation methods, have significantly expanded the range of tractable problems. Such challenges arise in problems with interfaces exhibiting high curvature, periodic structures, or near-contact between distinct components. This minisymposium will bring together researchers across these areas to highlight common challenges and emerging high-order integral equation methods for interface problems.

This minisymposium focuses on recent advances in the mathematical modelling and numerical simulation of poromechanics, which describes the coupled interaction between deformable porous media and fluid flow. Such models are fundamental to applications in geomechanics, biomechanics, groundwater flow, geothermal energy, hydrogen storage, petroleum engineering, and carbon capture and storage. The session will highlight developments in robust and efficient high order numerical methods for solving the underlying multiphysics partial differential equations, including mixed finite element, discontinuous Galerkin, finite volume, virtual element, hybrid high-order, and structure-preserving discretizations. Particular emphasis will be placed on adaptive algorithms, multiscale methods, efficient iterative solvers, error estimation, uncertainty quantification, and emerging machine learning approaches. The aim of this minisymposium is to bring together researchers working on the mathematical analysis, numerical approximation, and scientific computing aspects of poromechanics. Particular emphasis will be placed on the design, analysis, and application of high-order discretizations for coupled porous media problems.

This mini-symposium focuses on recent advances in the fundamentals of spectral methods, including new transforms for univariate and multivariate orthogonal polynomials, fast matrix decompositions and structured algorithms for systems arising from spectral discretisation, methods for fractional-order equations, approximation theory for spectral expansions, and novel spectral methods on non-standard and unbounded domains. Further topics include spectral approximation of singular functions, scaling strategies for Hermite and Laguerre methods, sparse spectral methods on geometries described by algebraic curves and surfaces, and stable high-order time-stepping schemes. The goal is to provide a platform for exchanging ideas, identifying common analytical and computational challenges, fostering collaboration, and advancing the state of the art in spectral methods.

This minisymposium focuses on recent advances in efficient and high-order numerical methods for atmospheric applications, e.g., in dynamical cores used for numerical weather prediction or climate modeling. Typically, the governing equations are the compressible Euler equations on the sphere or related models. Topics of interest include efficient time integration methods (e.g., horizontally-explicit and vertically-implicit (HEVI) methods), structure-preserving discretizations (e.g., energy-conserving, entropy-stable, well-balanced, asymptotic-preserving methods), robustness (e.g., bounds/positivity-preserving methods and schemes able to handle under-resolved flows), subgrid scale modeling for these methods, adaptivity (of models or in space/time), and the design of efficient and scalable (open-source) software. This minisymposium is intended to bring together researchers with different backgrounds, e.g., numerical analysis of compressible flows, high-performance scientific computing, and atmospheric sciences to foster discussions and collaborations.

High-dimensional problems are ubiquitous in scientific computing, arising in kinetic theory, quantum physics, multiscale flow and transport, and inverse problems, among others. Theories and numerical algorithms faces fundamental challenges due to the curse of dimensionality, demanding novel algorithmic frameworks and rigorous mathematical foundations. This minisymposium brings together researchers to present recent advances in structure-exploiting numerical methods for high-dimensional problems, spanning low-rank tensor approximations, model order reduction, multiscale techniques, AI-based surrogate modeling and PDE theories. These contributions highlight emerging algorithmic frameworks and mathematical theories that overcome the computational barriers of the curse of dimensionality across diverse scientific applications.

Complex nonlinear systems, governed by partial differential equations, are ubiquitous across science and engineering from computational fluid dynamics and plasma physics to phase-field modeling and biological pattern formation. While high-order numerical methods offer superior accuracy per degree of freedom, their practical deployment for such systems faces persistent challenges: maintaining provable stability and physical fidelity (e.g., entropy stability, positivity preservation, energy conservation/dissipation), ensuring computational efficiency on modern architectures, and handling complex geometries with unstructured meshes . This mini-symposium brings together recent advances in the development of high-order accurate numerical schemes that address these challenges through rigorous mathematical design and algorithmic innovation.

Solving wave propagation problems plays a crucial role in many industrial fields such as telecommunications, imaging techniques, seismology, non-destructive testing and radar-cross section computation. Addressing such a wide range of applications requires accurate, fast and practical simulation tools. Wave propagation phenomena are often modeled with boundary integral equation methods, whose discretizations are known as Boundary Element Methods (BEM). By requiring only the boundary of the physical domain to be meshed, these methods simplify the simulation setup and reduce the number of degrees of freedom compared with volume-based discretization approaches. However, BEM gives rise to dense linear systems, resulting in substantial memory requirements and computational costs that can be prohibitive for industrial-scale applications. Therefore, significant opportunities remain for advancing BEM methodologies to address the associated computational challenges. One research direction is the development of high-order discretization schemes to reduce the size of the linear system to be solved. The main goal of the Mini-symposium is to bring together experts and young researchers in the fields of numerical analysis and scientific computing, from both academic and industrial research groups, to discuss the most recent advances and current open challenges in innovative high-order BEM strategies. Application areas covered include, but are not limited to, elastodynamics, acoustics and electromagnetism. The Mini-symposium will focus on high-order BEM, both recent and traditional ones: h-, p- and hp-BEM, Isogeometric Analysis BEM (IgA-BEM), Virtual element approximation-based BEM (V-BEM), spectral BEM, etc.

Nonlocal models offer a robust framework for understanding and addressing complex real-world challenges where classical theories prove inadequate, particularly in the study of materials and structures characterized by discontinuities, singularities, and long-range interactions. Peridynamics, a reformulation of continuum mechanics based on integral equations, offers a mathematically sound framework that avoids spatial derivatives and naturally enables the modeling of crack initiation, propagation, and other discontinuous phenomena. Capturing the dynamics of such systems poses significant computational and analytical challenges due to their nonlocality and inherently multiscale nature. In addition to exhibiting discontinuous solutions, these problems are characterized by non-sparse stiffness matrices and singular convolution kernels. These intrinsic features lead to a significant increase in computational complexity and motivate the development of efficient and innovative solution strategies. This mini-symposium aims to address recent advances in both the theoretical and computational aspects of nonlocal methods, with a focus on applications to multiscale problems involving fracture and material discontinuities. The mini-symposium will bring together contributions spanning a broad range of methodologies and approaches, including, but not limited to, material failure, variational methods, robust numerical methods for complex structural analysis, finite and virtual element methods for crack simulation, multiscale modeling techniques, hybrid data-driven and physics-based approaches, reduced-order modeling and innovative techniques for uncertainty quantification and optimal control.

Spectral geometry concerns the interplay between the shape of a domain or manifold and the spectrum of an associated operator, such as the Laplace–Beltrami operator or the Dirichlet-to-Neumann (Steklov) map. In case of Extremal questions — which shapes optimize a given eigenvalue — connect analysis, geometry, and computation, with recent applications to free boundary minimal surfaces, sharp spectral inequalities, and computer-assisted proofs. Progress on these problems is often challenged by discretization accuracy: eigenvalue optimization demands many eigenpairs computed to high precision on evolving, non-smooth, or high-genus geometries. If the operators are vectorial or if the domain is unbounded, these questions present additional challenges. High-order methods — boundary integral and layer-potential formulations, the method of particular solutions and related fundamental solution approaches, spectral schemes, hp- and isogeometric finite elements — are uniquely suited to deliver, and increasingly certify, this accuracy. This minisymposium brings together researchers developing high-order discretizations for spectral problems in challenging settings with those applying them to optimization and inverse problems in spectral geometry.

High-order methods can provide high accuracy per degree of freedom and high arithmetic intensity that is well suited for modern computing architectures. Their adoption in industrial and large-scale simulations of partial differential equations (PDEs), however, remains limited by robustness issues for nonlinear, under-resolved, or shock-dominated problems and also by complexity of implementation. Recent developments in methods with provable properties, such as nonlinear stability, conservation, positivity preservation, and well-balancedness, provide a route to reliability. However, these properties often come at the expense of significant increase in computational cost. For these methods to become practical, such properties must be obtained without losing the efficiency that makes high-order methods attractive. The objective of this minisymposium is to bring together researchers addressing these challenges from complementary directions. Topics include entropy- and energy-stable schemes, positivity-preserving discretizations, limiting, shock capturing, adaptation, and performance-oriented implementations on modern computing platforms. We also seek contributions on high-order operator construction aimed at increasing accuracy per degree of freedom. These include construction of summation-by-parts (SBP) operators, e.g., Pad\'e-type, general-function space, and upwind SBP operators. Constructions that leverage structures such as collocation, sparsity, and tensor-product are also of interest. The overarching goal of the minisymposium is to clarify the trade-offs between accuracy, robustness, operator complexity, discretization cost, and hardware efficiency, and to identify practical directions for deploying robust and efficient high-order methods in industrial and large-scale PDE simulations.

Kernel methods have become ubiquitous across science and engineering. For more than two decades, radial basis function finite difference (RBF-FD) methods have provided flexible, high-order discretizations for partial differential equations on complex domains and manifolds. More recently, Gaussian Process (GP) perspectives have opened new directions in nonlinear solvers, uncertainty quantification, and operator learning, while retaining the potential for very high-order convergence. This four-slot minisymposium brings together researchers from these complementary communities to exchange ideas, identify common mathematical structure, and develop new connections between kernel-based numerical methods and kernel-based learning.

Over the past four decades, Chi-Wang Shu has profoundly shaped the field of computational mathematics through pioneering contributions to high-order numerical methods, hyperbolic conservation laws, discontinuous Galerkin methods, strong stability preserving time discretizations, and scientific computing. In celebration of his 70th birthday, this minisymposium brings together distinguished researchers whose work reflects the breadth and continuing impact of his scientific legacy. The presentations highlight recent advances in numerical methods for partial differential equations, including wave propagation, multiscale computation, adaptive algorithms for hyperbolic conservation laws, high-order time integration, and the mathematical analysis of numerical stability and accuracy. Together, these talks showcase both the enduring influence of Professor Shu’s ideas and the exciting new directions they continue to inspire across computational mathematics.

Time integration plays a fundamental role in the numerical simulation of complex multiscale and multiphysics systems, where disparate temporal scales and tightly coupled physical processes challenge existing algorithms. This minisymposium brings together researchers to discuss the design, analysis, and application of modern time discretization techniques. Presentations will span theoretical analysis, algorithmic development, and practical implementations, exploring topics that include high-order Runge-Kutta methods, geometric and structure-preserving integrators, adaptive and multirate time-stepping, exponential and IMEX methods, and time integration for coupled multiscale and multiphysics problems. By highlighting both methodological innovations and application-driven developments, the session seeks to promote interdisciplinary collaboration and advance the state of the art in time integration.

The accurate and efficient numerical simulation of partial differential equations requires methods that remain robust in the presence of nonlinearity, multiple scales, degeneracy, and strongly convective behavior. At the same time, practical numerical algorithms must preserve essential mathematical or physical structure while keeping computational costs manageable. This minisymposium brings together recent advances in time integration, novel spatial discretizations, space--time formulations, and model order reduction for computational PDEs. The presentations will address a variety of approaches both in time and space. These include perturbed Runge--Kutta methods and the development of their order conditions, smooth perturbations designed to improve the stability and efficiency of Runge--Kutta schemes, kernel-based tensor calculus in a unified space--time formulation, new discontinuous Galerkin methods, doubly conservative schemes for degenerate convection--diffusion equations, and model order reduction for convection-dominated problems. Together, the contributions illustrate how new analytical ideas and computational techniques can improve the accuracy, robustness, efficiency, and scalability of numerical methods for challenging PDE applications.

Structure-preserving Galerkin methods have emerged as a powerful class of numerical discretizations that faithfully reproduce key mathematical and physical properties of partial differential equations at the discrete level. By preserving structures such as divergence constraints, exact sequences, energy dissipation, symmetries, and long-time dynamics, these methods often deliver enhanced robustness, stability, and physical fidelity. This minisymposium aims to bring together researchers working on the design, analysis, and implementation of structure-preserving Galerkin methods. Topics of interest include compatible finite element methods, discontinuous Galerkin methods, hybridizable discontinuous Galerkin methods, hybrid high-order methods, mixed finite element methods, pressure-robust and divergence-conforming discretizations and adaptive algorithms. Applications include incompressible flow problems subject to divergence constraints, Maxwell equations, elasticity, wave propagation, kinetic equations, and other multiphysics systems. Particular emphasis will be placed on recent advances in preserving fundamental analytical structures, including mass conservation, incompressibility, energy and entropy stability, and hypocoercivity, as well as on rigorous numerical analysis, efficient solvers, and adaptive methodologies.

Scientific machine learning (SciML) has emerged as a promising framework for solving differential equations, inverse problems, multiscale models, and high-dimensional parametric systems. However, many SciML methods still face fundamental challenges in accuracy, stability, convergence, generalization, and computational efficiency, especially when data are sparse, noisy, or expensive to generate. This minisymposium will examine how numerical analysis, with particular emphasis on spectral and high-order methods, can guide the development of reliable and mathematically grounded SciML approaches. High-order finite element, spectral element, and discontinuous Galerkin methods provide accurate approximation spaces, structure-preserving discretizations, and rigorous tools for stability and error analysis. These principles can inform the design of network architectures, loss functions, training strategies, and validation procedures. Conversely, machine learning can enhance high-order simulations through operator learning, reduced-order modeling, adaptive discretization, closure modeling, preconditioning, and solver acceleration. The minisymposium will emphasize hybrid approaches that integrate data-driven models with numerical discretizations and governing physical laws rather than treating machine learning as a replacement for classical methods. Particular attention will be given to robustness across mesh resolutions and polynomial orders, conservation, and scalable implementation. Topics include physics-informed neural networks, neural operators, structure-preserving learning, inverse problems, uncertainty quantification, data assimilation, and applications to fluid mechanics and related scientific computing problems.

Numerical methods for Partial Differential Equations based on general polygonal and polyhedral meshes represent a highly active and evolving research field. Their success is primarily driven by their inherent flexibility in handling complex geometries, which supports seamless adaptive mesh refinement and enables the construction of discrete spaces with tailored physical or mathematical properties. This minisymposium aims to mix rigorous mathematical foundations and high-per\-for\-man\-ce computational applications. We welcome contributions on a wide range of polytopal frameworks, such as the Virtual Element Method, Hybrid High-Order methods, and Discontinuous Galerkin schemes, focusing on both their theoretical properties and their impact on complex simulations. Key topics include advanced error analysis (a priori and a posteriori), stability, and the preservation of physical structures (e.g., divergence-free constraints or tensor symmetry) at the discrete level. Simultaneously, the session will explore the practical challenges arising in computational mechanics, fluid dynamics, and multiphysics. Particular interest is also devoted to the design of efficient preconditioners, scalable solvers, and parallel implementations, which are essential for transitioning these modern discretizations from theoretical prototypes to robust tools for large-scale scientific computing.

The industrialisation of high-order methods is already well under way within major aeronautical companies such as Airbus and Rolls-Royce. Advances in GPU-based and heterogeneous high-performance computing, together with efficient matrix-free implementations, are making high-order finite and spectral element methods increasingly attractive for large-scale PDE simulations, enabling highly accurate solutions that can, in some cases, approach the DNS range. Their superior accuracy and convergence properties are well established; however, practical applications, particularly those involving transport-dominated or nonlinear problems, often require stabilisation, limiting, or other algorithmic modifications to ensure robustness. Because these enhancements can alter the formal properties of the underlying discretisation, an important question remains: under what conditions are high-order methods truly practical, and when do they provide a meaningful advantage over lower-order schemes? This minisymposium brings together researchers developing and applying high-order discretizations across a range of computational science and engineering applications to address this question from different perspectives. Topics include structure-preserving discretizations (e.g., energy-conserving and entropy-stable methods), stabilization strategies, positivity- and bounds-preserving schemes, and robust solver technologies. Applications span weather and climate modeling, compressible flows, turbomachinery, wave propagation, and other large-scale multiphysics simulations. By highlighting common challenges and successful strategies across disciplines, this minisymposium aims to identify the algorithmic and computational principles that make high-order methods practical for production-scale scientific computing.

High-order computational fluid dynamics (CFD) solvers are increasingly important for high-fidelity simulations in science and engineering. However, their widespread adoption for large-scale applications remains limited by computational cost, scalability, and robustness challenges. This minisymposium will highlight recent developments in high-order CFD solvers that improve accuracy, efficiency, and stability while reducing overall time-to-solution. Particular emphasis will be placed on advances that leverage both discretization and solver technologies to reduce computational cost without compromising predictive capability. A central theme of the minisymposium is the development of spatial and temporal discretization strategies that enable scalable solvers on modern high-performance computing architectures. These approaches play a critical role in improving solver robustness, reducing iteration counts, enhancing parallel efficiency, and enabling efficient simulations at unprecedented scales. Topics include high-order discretizations for compressible and incompressible flows, adaptive methods, space-time formulations, and techniques that improve the conditioning and performance of large-scale solver frameworks. Contributions addressing provably stable discretizations, shock-capturing methodologies, and advanced time-integration schemes are also encouraged. The minisymposium will further explore the interaction between discretization design and linear and nonlinear solver performance, including advances in preconditioning and solver acceleration strategies. Applications demonstrating practical benefits for industrially relevant CFD problems are particularly welcome. Overall, the objective is to showcase developments that are making high-order CFD solvers more accurate, scalable, stable, and computationally affordable for the next-generation of predictive simulations.

Hyperbolic PDEs govern a wide range of models in geophysics and continuum mechanics, from shallow water flows to compressible gas dynamics, multiphase flows, and elasticity. Their numerical solution requires high-order schemes that remain robust in the presence of shocks, moving interfaces, and stiff source terms, while preserving, at the discrete level, the structural properties of the continuous models, such as entropy inequalities, positivity of some variables, etc. This minisymposium brings together recent developments on discontinuous Galerkin, finite volume, and finite element methods addressing these issues, including entropy-stable and invariant-domain-preserving discretizations, well-balanced schemes, all-Mach formulations, subcell finite volume techniques, shock capturing on fixed and moving meshes, and efficient implicit and semi-implicit time integration. The applications considered range from shallow-water, landslide, and thin-film models, to multiphase systems and hyperbolic reformulations of continuum mechanics. The contributions span the entire spectrum, from the construction and analysis of the schemes to their implementation and validation through demanding benchmarks.

Nonlinear hyperbolic balance laws serve as the mathematical foundation for a vast range of fluid dynamic processes. Despite their widespread use, addressing fundamental hyperbolic phenomena, such as the complex multi-scale interaction of shock waves with shear instabilities, or the coupling of hyperbolic transport with stochastic environments, remains a challenge for numerical modeling. A prominent example is the inadequacy of the classical entropy principle to guarantee unique solutions in continuum-scale systems, often giving rise to highly oscillatory, non-unique flow fields. Consequently, this minisymposium aims to explore recent breakthroughs and novel directions in the design of tailored, robust numerical frameworks for hyperbolic balance laws. We welcome contributions focusing on innovative solution concepts, multi-scale methodologies, and computational approaches for probabilistic fluid dynamics. This scope encompasses high-resolution and high-order discretizations exploiting generalized entropy formulations, dissipative limits, or advanced approaches for non-local and multi-scale problems, such as subgrid-scale modeling within continuum frameworks. Furthermore, emphasis is placed on structure-preserving and stable schemes, including recent advances in entropy-conservative or entropy-stable methods, well-balancing, and the accurate capturing of asymptotic states like low-Mach limits. Finally, the session targets uncertainty quantification and algorithmic developments for stochastic hyperbolic systems, featuring probabilistic formulations for statistical turbulence via stochastic variational principles or hybrid data-driven and filtering techniques. This session serves as a collaborative platform highlighting ongoing research from the German Priority Research Program SPP-2410 (”Hyperbolic Balance Laws in Fluid Mechanics: Complexity, Scales, Randomness”), while warmly inviting the broader international community to contribute, exchange ideas, and discuss future trajectories in the numerical analysis and simulation of hyperbolic systems.

Spectral, spectral element, and hp finite element methods are central high-order tools for high-fidelity scientific computing. Their high accuracy, tensor-product structure, locality, and high computation-to-memory ratio make them well suited for modern heterogeneous and exascale architectures. Beyond accuracy, however, reliable simulations of complex physical systems require the preservation of intrinsic structures such as conservation laws, divergence constraints, positivity, entropy dissipation, symplecticity, and energy stability. The rise of AI for Science offers new opportunities to accelerate and augment high-order methods through neural operators, physics-informed learning, learned solvers, data-driven closures, and AI-assisted adaptivity. Yet robustness, physical consistency, and long-time stability remain major challenges for purely data-driven models. This minisymposium focuses on structure-preserving and high-fidelity spectral/high-order methods in the era of AI4S, emphasizing rigorous numerical foundations, efficient HPC implementations, and principled integration with machine learning. Topics include conservative, divergence-free, positivity-preserving, entropy-stable and energy-stable schemes; asymptotic-preserving and symplectic methods; fast solvers, preconditioners, and GPU-oriented implementations; spectral or polynomial-basis neural operators; variational and physics-informed learning; learned hp-adaptivity; AI-assisted preconditioning; and structure-preserving neural closures, with applications in fluids, electromagnetics, kinetic equations, waves, materials, geophysical flows, and multiphysics simulations.

High-fidelity computational fluid dynamics is a key tool for the simulation, analysis, and control of complex flows. However, the associated computational cost remains prohibitive in many-query settings, including parametric studies, uncertainty quantification, inverse problems, optimization, and real-time prediction. Moving from high-fidelity simulations to fast predictive models requires methods that preserve the essential physical and numerical features of the underlying flow dynamics while substantially reducing the computational burden. This minisymposium focuses on recent advances in reduced-order modeling and scientific machine learning for CFD, with particular emphasis on methods that combine physical structure, numerical accuracy, and data-driven approximation to build fast and reliable surrogate models. Contributions addressing both methodological developments and application-driven challenges are welcome. Topics include projection-based and non-intrusive reduced-order models, reduced basis and POD techniques, empirical interpolation and hyper-reduction, operator learning, physics-informed and structure-preserving learning, neural surrogates for parametric and time-dependent flows, data assimilation, uncertainty quantification, optimization, and control. Particular attention will be given to the interplay between ROM, SciML, and high-fidelity numerical solvers, including spectral, high-order, discontinuous Galerkin, and adaptive methods, as well as to applications in realistic flow configurations. Examples of interest include, but are not limited to, turbulent and multiphysics flows, fluid--structure interaction, biomedical flows, environmental and geophysical flows, and industrial or engineering design problems. The minisymposium aims to promote discussion on accuracy, stability, generalization, robustness, and efficiency, with a view toward the reliable use of reduced and data-driven models in real-world CFD applications.

High-dimensional functions and operators arise throughout science and engineering, yet their approximation is hindered by the curse of dimensionality. Structured representations such as tensor networks, hierarchical decompositions, and compositional models (including feed-forward neural networks) can exploit the underlying structure. However, their practical performance depends critically on training sample selection, making active learning and optimal sampling essential for efficient approximation. This mini-symposium brings together researchers working on approximation theory, tensor methods, compositional models, and adaptive sampling to discuss recent advances in structured approximation and sample-efficient learning. Topics include sampling strategies, error estimates, anisotropic and hierarchical sparsity, and efficient algorithms for high-dimensional approximation.

Pressure-robust finite element methods have emerged as a powerful class of discretizations for incompressible flow problems. In recent years, a variety of such methods have been developed and analyzed, many of them being high-order. Their defining feature is that the velocity error is independent of the pressure, making them particularly effective in situations where the pressure exhibits low regularity. This structure-preserving property is especially important for time-dependent problems, where it is related to long-time stability and allows for viscosity-robust error estimates. Moreover, in coupled systems involving nonlinear terms that generate gradient fields, pressure-robust methods prevent spurious influences of these gradients on the discrete velocity, providing a significant advantage over standard discretizations. This minisymposium brings together recent advances in the numerical analysis of pressure-robust finite element methods, including H1-conforming exactly divergence-free finite elements, H(div)-conforming methods, and reconstruction operator approaches. We will discuss their convergence, and conservation properties, as well as their computational costs and implementation aspects. By comparing different techniques, the minisymposium aims to provide a clearer understanding of the trade-offs between accuracy and robustness, and to identify the regimes and applications in which pressure-robust methods offer the greatest benefits. Ultimately, the goal is to facilitate their transfer to more advanced coupled fluid models and to challenging real-world applications.

Accurate and efficient simulation of problems in continuum mechanics relies heavily on advanced finite element methods and related numerical techniques capable of resolving complex solution structures. Recent advances in high-order and nonstandard finite element discretizations, together with emerging operator-learning approaches, have broadened the computational tools available for complex partial differential equations while helping preserve essential mathematical structures and physical properties. This minisymposium brings together researchers working on the theory, analysis, algorithms, and applications of these methods. Topics include mixed and hybridizable finite element methods, structure-preserving discretizations, and efficient iterative algorithms, with applications to eigenvalue and interface problems, solid and fluid mechanics, wave propagation, and related areas of computational science. The minisymposium will provide a forum for exchanging recent advances and fostering collaboration among the numerical analysis and computational mechanics communities.

Scientific Machine Learning (SciML) is an emerging scientific field at the interface of data science, high-dimensional approximation, statistics, and deep learning methodologies. Central themes are operator approximation and operator learning. This MS will present recent contributions on spectral, polynomial, and other "high-order'' approximation and computation methodologies for operator surrogates in SciML. Recent progress in spectral and polynomial approximation theoretical results and approximation algorithms will be reviewed, on the one hand in operator learning and on the other hand in deep learning and statistical learning theory and computation in relation to operator approximation.

Many partial differential equations arising in geometry, continuum mechanics, and mathematical physics involve unknowns constrained to nonlinear manifolds. Examples include geometric evolution equations, nonlinear elasticity, shell and rod models, Landau-Lifshitz dynamics, harmonic and wave maps, and theories of liquid crystals and other constrained materials. Their numerical approximation poses significant challenges due to the interplay between geometric constraints, nonlinearity, and stability. This minisymposium focuses on the development and analysis of higher-order numerical methods for manifold-valued PDEs. Topics include high-order finite element and discontinuous Galerkin methods, geometric and structure-preserving time integrators, error analysis, adaptive algorithms, efficient nonlinear solvers, and applications to continuum mechanics, geometric evolution equations, and constrained physical models. The aim is to bring together researchers working on the mathematical foundations, algorithmic development, and practical implementation of advanced discretization techniques for manifold-valued problems, fostering interactions between numerical analysis, scientific computing, and applications.

Recent years have witnessed the emergence of Scientific Machine Learning (SciML) as a new paradigm for the numerical treatment of differential equations. Complementing traditional discretization techniques, SciML leverages machine learning algorithms to address forward and inverse PDE problems, construct surrogate models, learn solution operators, and accelerate traditional numerical methods. Despite the remarkable success across a broad range of applications, many fundamental questions remain open regarding the mathematical foundations, convergence properties, computational efficiency, and generalization of these methods. Addressing these challenges naturally calls for concepts and techniques from high-order numerical methods. Variational formulations, accurate quadrature, approximation theory, adaptive sampling, high-order discretization schemes and rigorous error analysis offer powerful tools for improving both the theoretical understanding and practical performance of SciML algorithms. This minisymposium aims to bring together researchers working at the intersection of Scientific Machine Learning and Numerical Analysis to highlight open challenges, explore emerging synergies, and stimulate new research directions at the interface of these fields. We welcome contributions spanning theoretical analysis, algorithmic developments, and computational applications.

Fokker--Planck (FP) equations describe the evolution of probability densities under stochastic dynamics, while Mean-Field Games (MFG) couple such a forward (in time) equation with a backward Hamilton--Jacobi--Bellman equation to model the equilibrium behaviour of large populations of interacting agents. Both arise across statistical physics, biology, economics, and machine learning, and both are closely tied to the geometry of optimal transport and Wasserstein gradient flows. However, their numerical solution is hindered by the curse of dimensionality, so that traditional grid-based discretizations quickly become intractable. Structured methods such as low-rank tensor formats, sparse grids, neural networks, and particle approximations can exploit the underlying structure, making high-dimensional problems of practical interest computationally feasible. This mini-symposium aims to bring together researchers working on high-dimensional approximation for FP equations and MFGs. Topics include low-rank tensor and sparse-grid methods, neural-network and particle approximations, model-order reduction, and optimal-transport and gradient-flow schemes, alongside the numerical analysis that supports them. A further goal is to connect the FP and MFG communities, which are mathematically closely related but often work separately.

This minisymposium focuses on state-of-the-art numerical methods for PDE eigenvalue problems, central in scientific computing due to their critical importance in stability analysis, vibration modeling, quantum physics, etc. Presentations will highlight novel discretization and approximation techniques for standard and nonstandard PDE eigensolvers, as well as approaches for nonlinear eigenvalue problems. Some time will also be dedicated to talks on outer-loop methodologies designed to handle parametric and stochastic eigenvalue frameworks such as those arising in inverse problems, optimal design, and uncertainty quantification.

Evolutionary Partial Differential Equations (PDEs) are of primary relevance: they arise pervasively in nature and, consequently, they are present in a wide range of problems in engineering and applied physics. The numerical approximation of their solutions is a challenging task that has always attracted interest within the scientific community. This minisymposium is devoted to novel and advanced high-order space--time methods for computing accurate and efficient numerical approximations of the solutions of evolutionary PDEs. Its main goal is to bring together experts in numerical analysis from different international research groups to discuss the most recent advances and the open challenges. Topics of interest include, but are not limited to, conforming discretizations, least-squares formulations, $hp$-convergence, a priori and a posteriori error estimates, stability, structure-preserving methods, optimal control, and isogeometric discretizations, with applications to parabolic, hyperbolic, and dispersive problems.

High-order methods are attractive for the numerical solution of partial differential equations, offering accurate discretization of curvilinear geometries at a lower computational cost than low-order approaches for a given accuracy. Despite this promise, high-order mesh generation remains a significant bottleneck, particularly for complex, real-world configurations. Problems involving transient geometry and/or solution features add a further layer of difficulty, since no single static mesh can efficiently resolve the full range of scales and features encountered throughout a simulation. High-order mesh adaptation, through h-, r- and p-adaptivity, and combinations thereof, offers a route to accurate, efficient solutions without the cost of repeatedly regenerating meshes from scratch. This minisymposium provides a forum to discuss recent advances in high-order mesh generation and adaptation, bringing together leading academic and industrial experts. Topics of interest include, but are not limited to: geometry handling and CAD integration for high-order workflows; curvilinear mesh generation and validity/untangling techniques; the use of machine learning to assist in high-order meshing; mesh curving strategies for complex and industrial-scale configurations; boundary layer and hybrid mesh generation; anisotropic and metric-based adaptation; error estimation to drive adaptive strategies; r-adaptivity and mesh deformation for moving/deforming domains; hp-adaptivity and its integration with high-order solvers. Last, applications of high-order meshing and adaptation algorithms for large-scale, parallel or industrial test-cases are encouraged.

This workshop addresses current frontiers in the numerical analysis of nonlinear partial differential equations (PDEs). The construction of efficient, reliable, and robust numerical methods for nonlinear systems is an active area of research. Applications span a wide range of models in electromagnetism, mechanics, fluid dynamics, and general relativity, including systems coupled with optimal control problems and geometric PDEs. Within this broad field, the workshop will emphasize nonlinear PDEs governed by algebraic and geometric structures and examine how these structures guide the development of computational methods. Nonlinear PDEs involving vector and tensor fields are a central theme of the workshop. The numerical analysis of scalar-valued nonlinear PDEs has seen considerable progress in recent years. PDEs arising in vector and tensor analysis exhibit additional structures, such as compatibility conditions and geometric invariance, that are less prominent in the scalar-valued setting. Computational methods for linear PDEs treat these structures as foundational, and they are equally fundamental to the development of computational methods for nonlinear problems. Many nonlinear systems arise from physical models coupled with evolving geometries. The workshop will therefore emphasize connections with nonlinear and geometric PDEs on surfaces and manifolds. This workshop will be an opportunity to explore links between these research efforts. We intend to invite speakers who address the theoretical analysis, the construction of discretizations, and computational algorithms. The broad scope of the program will connect complementary methodologies actively researched in the numerical analysis of nonlinear partial differential equations, including mixed finite element methods, finite volume methods, and polytopal schemes. The program will address stationary elliptic problems and time-dependent problems of parabolic and hyperbolic nature. This workshop will contribute to ICOSAHOM 2027 by stimulating the exchange between communities that often address closely related structural questions using different mathematical languages.

This minisymposium focuses on modern high-order techniques for the numerical simulations of geophysical and environmental flows. High-order methods have shown great potential for improving the numerical simulation of the compressible or incompressible Euler and Navier–Stokes equations, Boussinesq-type equations, and the shallow water equations, across various geophysical regimes. The minisymposium will cover high-order spatial discretization techniques in the horizontal and vertical directions (e.g., discontinuous Galerkin, semi-Lagrangian, high-order finite volume, spectral element methods, multilayer-moment models) as well as high-order, efficient and stable time integration strategies for multiscale geophysical problems and novel structure-preserving discretizations (e.g., well-balanced, equilibria-preserving, positivity-preserving, energy-conserving, entropy-stable, and asymptotic-preserving schemes). The minisymposium will address the many challenges associated with the application of high-order methods to environmental flows. In particular, while high-order methods implemented on the same mesh as low-order methods remain computationally expensive and therefore require scalable parallel solvers and modular implementations, their use on coarser meshes introduces different challenges, such as representing unresolved flow scales or sub-grid topography/bathymetry scales. Applications include, but are not limited to, coastal, oceanic, and atmospheric flows, as well as coupled problems involving moist advection, cloud resolution, sediment and plastic transport. The goal of this minisymposium is to create a venue where methods developed independently for different environmental applications can be compared and cross-fertilized, encouraging speakers to highlight not only their results but also the modeling assumptions and discretization choices behind them, so as to stimulate discussion on which techniques generalize across the different flow regimes represented.

This mini-symposium presents recent developments in advanced spline methods for Isogeometric Analysis (IGA). It brings together theoretical and computational perspectives on the construction and analysis of spline spaces, with a focus on approximation, adaptivity, efficiency, and flexibility. Particular attention is given to modern spline technologies that enable refined and adaptive discretizations, improved computational performance, and the treatment of complex geometries and challenging problems. The mini-symposium aims to provide an overview of current trends in spline-based IGA and to foster discussion on new developments, challenges, and future directions in the field.

Isogeometric analysis provides a high-order framework for the numerical solution of PDEs. Originally motivated by the integration of numerical simulation with CAD-based geometry, it has shown significant potential as a flexible methodology based on smooth high-degree spline spaces. At the same time, this methodology raises significant theoretical and computational challenges, including structure-preserving discretizations, smooth spline spaces on triangulations and unstructured meshes, immersed methods for complex geometries, local refinement techniques, efficient operator formation, and robust solvers for the resulting linear systems. This minisymposium aims to present and discuss recent advances in these areas. The minisymposium also aims to foster interaction between the isogeometric analysis community and the broader spectral and high-order methods community.

Adaptive high-order methods provide a powerful framework for the accurate and efficient numerical approximation of partial differential equations and associated eigenvalue problems, particularly in the presence of singularities, localized features, complex geometries, or multiple spatial scales. This minisymposium aims to bring together young researchers working on theoretical, computational, and practical aspects of adaptive numerical methods across a broad range of approximation frameworks, including finite element methods, space–time approaches, and other high-order techniques. Particular emphasis will be placed on a posteriori error estimation, reliability and efficiency analysis, convergence and optimality of adaptive algorithms, and mesh and polynomial degree refinement strategies. By presenting complementary approaches and recent developments, the minisymposium seeks to promote scientific exchange among early-career researchers, strengthen existing scientific connections, and foster new collaborations.

High-order discretizations provide exceptional accuracy, but their computational efficiency depends critically on solvers that exploit local structure while controlling the cost of global coupling. Substructuring and hybridization address this challenge by organizing the computation around local problems coupled through interfaces, traces, or skeleton variables. Depending on the formulation, these interface unknowns arise from a decomposition of the discrete problem or are introduced as a part of the discretization itself. This minisymposium brings together research on the analysis, design, and implementation of solvers based on this local-to-global structure. It encompasses methods that eliminate or solve for interior unknowns locally and formulate the remaining global interaction as an interface problem. The emphasis is on the resulting solver architecture, whether the interface formulation is obtained algebraically, constructed through hybridization, or developed directly at the operator level. The minisymposium focuses on questions that become particularly important in the high-order regime: the construction and representation of interface operators, robustness with respect to discretization parameters and problem complexity, computational and memory efficiency, parallel scalability, and the interaction between local discretization structure and global solution strategies. Its aim is to connect researchers working with different formulations through the shared mathematical and computational principles of local elimination and interface coupling.

High-order discretizations provide a powerful foundation for fast, efficient, and accurate scientific simulations on exascale and emerging computing architectures. At the same time, artificial intelligence and scientific machine learning are creating new opportunities to accelerate simulation, enhance solution accuracy, construct scalable surrogate models, and enable data-driven discovery and optimization. This minisymposium will highlight advances at the intersection of high-order numerical methods, performance-portable software,exascale computing, and AI-enabled scientific computing. Topics will include finite and spectral element algorithms; high-order computational kernels and lightweight portable libraries; scalable solvers and preconditioners; performance portability across heterogeneous architectures; and exascale applications. The minisymposium will also feature AI and machine-learning approaches that complement high-fidelity simulation, including physics-informed learning, neural operators, reduced-order and surrogate models, super-resolution methods, foundation models, and AI-assisted simulation, design, and control.

Hyperbolic conservation laws underpin models throughout fluid dynamics, geophysics, and engineering, yet their solutions develop discontinuities that challenge classical numerical methods and modern data-driven approaches alike. Recent years have seen rapid progress in machine-learning-based techniques for these systems, including neural solvers and operator learning, surrogate models for many-query applications, and learning-augmented high-order discretizations. At the same time, discontinuous solutions, entropy conditions, and conservation constraints expose fundamental limitations of off-the-shelf learning methods, raising questions of accuracy, stability, and structure preservation that lie at the heart of the high-order methods community. This minisymposium will highlight recent advances in data-driven methods for hyperbolic conservation laws. Topics of interest include neural solvers and operator learning for systems with discontinuous solutions; machine-learning-augmented high-order methods; structure-preserving and entropy-stable learning approaches; Bayesian and deterministic inverse problems for hyperbolic systems; and data assimilation methods that respect conservation, invariants, and solution regularity. Particular emphasis will be placed on approaches that combine the rigor of high-order discretizations with the flexibility of data-driven techniques, including hybrid solver-informed strategies and uncertainty quantification. The minisymposium aims to bring together researchers from numerical analysis, scientific machine learning, inverse problems, and data assimilation to discuss theoretical foundations, algorithms, and emerging applications, fostering exchange between the high-order methods and scientific machine learning communities.

The brain is a highly complex, multiphysics system in which electrical, chemical, and mechanical processes interact across a wide range of spatial and temporal scales. Accurate computational modeling of such phenomena requires not only sophisticated physical descriptions, but also advanced numerical methods capable of resolving multiscale features and strong couplings with high fidelity and efficiency. This minisymposium brings together researchers developing and applying advanced numerical methods for multiscale and multiphysics brain modeling. Topics of interest include high-order finite element and discontinuous Galerkin methods, polygonal/polyhedral and agglomerated discretizations, isogeometric analysis, robust and scalable solvers for coupled systems, adaptivity (p-, h-, and hp-refinement), error estimation and control, and numerical methods for complex geometries and heterogeneous biological tissues. Relevant application areas include the mechanics of brain tissue and cerebrospinal and interstitial fluids, transport and clearance processes (including glymphatic and perivascular pathways), electromechanical coupling, prion-like protein spreading, tissue atrophy, electrophysiology and network-scale dynamics. Contributions integrating physics-based modeling with data-driven approaches, including reduced-order modeling and data assimilation from imaging or clinical datasets, are especially encouraged.

Many variational problems in science and engineering are governed by constraints that encode essential physics. Equality constraints, such as incompressibility, and conservation of energy or momentum, underpin structure-preserving discretizations. Inequality constraints are equally pervasive, arising in free boundary problems, ice-sheet flow, groundwater filtration, and mathematical finance, as well as in regularization-based shock capturing for hyperbolic conservation laws. A unifying perspective is the geometric treatment of the structure governing the underlying physics, spanning conservation laws, exact sequences, and the nonsmoothness of contact, obstacles, and shocks. Designing high-order schemes that preserve structure, or resolve sharp features, with mesh-independent convergence remains an active research area. This minisymposium brings together researchers at the forefront of algorithmic and theoretical developments in physics- and structure-preserving numerical methods for variational problems, with an emphasis on: (a) compatible discretizations, optimization algorithms to achieve mesh independence, structure and constraint preservation, and sharp-feature resolution; (b) a priori and a posteriori error estimates; and (c) applications in contact mechanics, geophysics, electromagnetics, topology optimization, compressible fluid dynamics, and other areas. The session aims to identify open problems including adaptive high-order methods, efficient solvers for large-scale constrained problems, and the coupling of constrained and regularized variational formulations with other physical models.

The computational modeling of complex physical systems in sustainable design, healthcare, and engineering requires accurate, efficient, and problem-specific predictive methods. Standard numerical discretizations often struggle to preserve key physical invariants and mathematical structures, leading to unphysical approximations that compromise reliability. This minisymposium aims to bring together experts in advanced numerical strategies that overcome these issues through structure-preserving and physically consistent methods. The session spans three main methodological frameworks. First, we explore high-order structure-preserving discretizations -- such as polytopal discontinuous Galerkin, virtual element, and Trefftz methods -- tailored for complex dynamics, including incompressible and magnetohydrodynamic flows, fractured media, and wave propagation. Second, we discuss structure-preserving model order reduction for parametrized PDEs, including hyper-reduction, neural network-based nonlinear approximations, and closure models designed for transport-dominated or shock-driven phenomena. Third, we aim at combining these methods with data integration and data assimilation to preserve physical consistency in the context of noisy and sparse data.

For many important applications, the underlying partial differential equations are not uniformly coercive. These include problems with indefinite or non-selfadjoint operators and stability estimates that depend on physical parameters. % Discrete stability may depend on the mesh. Examples occur in wave propagation, convection-dominated problems, and problems with sign-changing coefficients. In such settings, standard Galerkin methods may fail to preserve the stability of the continuous problem. Two complementary frameworks address this difficulty. Minimum-residual methods compute an approximation by minimizing the residual in a suitable dual norm. The choice of the test inner product, which determines the associated trial-to-test operator, is crucial for the stability of the method and the quality of the approximation. For linear problems in Hilbert spaces, residual minimization can yield symmetric positive-definite formulations even when the original operator is indefinite. % The computable residuals provide a posteriori error estimators that can be used for adaptive mesh refinement. Classically, the well-posedness of variational forms of linear problems is achieved thanks to the inf-sup condition and the solvability condition. These conditions can be derived with the help of T-coercivity. A bounded, invertible transformation operator is introduced so that the transformed bilinear form becomes coercive (possibly up to a compact perturbation). This makes it possible to establish an inf-sup condition or apply Fredholm theory when the original form is not coercive. The transformed problem is equivalent to the original one and stability estimates can be transferred between the two formulations. Both theories share a common principle: they associate trial functions with suitable test functions to establish stability. Minimum-residual methods construct this association through the trial-to-test operator, while T-coercivity uses a transformation that makes the variational form coercive. This raises the common question: how can these operators be approximated while preserving stability at the discrete level? This minisymposium will bring together experts in both minimum-residual methods and T-coercivity to present their latest research and discuss the connections between these two perspectives. Exploring this connection may allow analytical tools and computational techniques developed in either framework to benefit the other.

Data-driven methods (including reduced order models, machine-learning surrogates, or data-assimilation techniques) for physics-based simulations have demonstrated remarkable utility and effectiveness; however, achieving robust performance in extrapolatory and predictive regimes often remains challenging. Embedding physical structure-preserving properties (e.g., balance laws, preservation of geometric or solution invariants) can significantly improve performance in these regimes. These techniques also improve performance in the presence of challenging fluid physics, including shock waves, boundary layers, and turbulence. This minisymposium focuses on techniques for enforcing such properties within data-driven methods, which can both generalize and deviate significantly from approaches used in traditional numerical methods.

Imbuing numerical algorithms with physical structure-preserving properties has led to a renaissance in physics-based simulations. These properties include, but are not limited to, conservation and dissipation of auxiliary quantities, preservation of key steady states or solution invariants. Mimicking such properties significantly improves the robustness and accuracy of simulations in a wide range of applications in science and engineering. This minisymposium focuses on the application of emerging structure-preserving techniques within complex fluids, including (but not limited to) multi-species and multi-phase flows, thermally perfect gases, and non-ideal equations of state.

Many variational problems in continuum mechanics, including the Hellinger–Reissner formulation of linear elasticity, the velocity–stress formulations of incompressible and complex flows, and their magnetohydrodynamic and liquid-crystalline counterparts, are governed by a stress tensor whose algebraic structure is dictated by the balance of angular momentum. In the classical setting this manifests as the symmetry of the Cauchy stress, while in polar and micropolar media it manifests as a controlled non-symmetry. Recent work has shown that whether a discretization respects this structure exactly, rather than approximately, may drastically affect the accuracy of the computed stress: schemes enforcing the structure only weakly may return arbitrarily poor stress approximations under anisotropic constitutive laws, even in zero-stress configurations, whereas schemes enforcing it strongly deliver stress errors independent of the constitutive law. We have termed this property material robustness, in analogy with pressure robustness in incompressible flow. Strongly symmetric elements often come in two flavors: very low-order, or impractically high-order with supersmoothness. Constructing intermediate, but still high-order, spaces and appropriate solvers remains an active area of research that some of the talks will address. Moreover, circumventing the symmetry constraint via weak imposition also requires high-order methods to obtain accurate solutions for some constitutive laws. This minisymposium gathers recent advances on material robustness across theory, implementation, methodology, and applications: the unifying theory and its finite element tensor calculus interpretation; high-order frameworks such as TDNNS and Trefftz methods, and the solvers they require; and applications to thin domains, liquid crystals, magnetohydrodynamic equilibria, and complex fluids. Together, the talks position structure preservation as a unifying design principle for accurate, reliable, high-order discretizations throughout continuum mechanics. We envisage two to three sessions.