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.