Geometric Structure-Preserving Finite Element Methods: Mathematical Theory and Applications
Keywords:
Structure-Preserving Methods, Finite Element Exterior Calculus, De Rham Complex, Symplectic Integration, Mixed Finite Elements, Hamiltonian Systems, Nédélec Elements, Computational MathematicsAbstract
Structure-preserving numerical methods have emerged as a fundamental paradigm in computational mathematics, ensuring that discrete approximations inherit essential geometric and physical properties of continuous systems. This study presents a comprehensive mathematical analysis of geometric structure-preserving finite element methods, encompassing finite element exterior calculus (FEEC), symplectic integration, and mixed formulations for saddle point problems. The theoretical framework develops the discrete de Rham complex using Nédélec edge elements, Raviart-Thomas face elements, and their higher-order extensions. We establish that commuting diagram properties guarantee stability of mixed discretizations through automatic satisfaction of the inf-sup condition with mesh-independent constant . For Hamiltonian systems, symplectic integrators preserving the two-form are analyzed, demonstrating bounded energy errors over arbitrary time intervals. Applications to Maxwell eigenvalue problems show complete elimination of spurious modes, incompressible Navier-Stokes simulations achieve pointwise , and wave propagation maintains discrete energy conservation . Optimal convergence rates are established within the FEEC framework, with improved estimates under regularity assumptions. Computational experiments validate all theoretical predictions and demonstrate the practical advantages of structure preservation for challenging multi-physics applications.
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