Numerical Stability and Accuracy of Structure-Preserving Finite Element Schemes
Keywords:
Numerical Stability, Finite Element Accuracy, Structure-Preserving Methods, Inf-Sup Condition, Convergence Analysis, Error Estimation, Conservation Laws, Computational EfficiencyAbstract
The numerical stability and accuracy of finite element methods are fundamental considerations determining the reliability and efficiency of computational simulations. This study presents a comprehensive analysis of stability and accuracy properties for structure-preserving finite element schemes based on finite element exterior calculus (FEEC). We establish rigorous stability results including coercivity estimates with constant , inf-sup conditions with mesh-independent constant , and energy preservation bounds for Hamiltonian systems. Accuracy analysis demonstrates optimal convergence rates and for degree- elements, with superconvergence at Gauss points. The framework encompasses elliptic, parabolic, and hyperbolic problems, with particular attention to saddle-point systems arising in incompressible flow and electromagnetics. Condition number analysis shows for standard formulations, reduced to with optimal preconditioning. Numerical experiments on benchmark problems—including the Stokes equations, Maxwell eigenvalue problems, and Hamiltonian systems—validate theoretical predictions and demonstrate the superior physical fidelity of structure-preserving approaches. We show that exact satisfaction of discrete constraints eliminates spurious modes, preserves conservation laws to machine precision, and enables long-time stability without artificial dissipation. Computational efficiency comparisons demonstrate that structure-preserving methods achieve target accuracy 20–40% faster than standard approaches despite increased per-element complexity.
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