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arXiv · 2501.11494

A variational approach to the analysis of the continuous space-time FEM for the wave equation

Sergio Gómez
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Abstract

We present a stability and convergence analysis of the space-time continuous finite element method for the Hamiltonian formulation of the wave equation. More precisely, we prove a continuous dependence of the discrete solution on the data in a $C^0([0, T]; X)$-type energy norm, which does not require any restriction on the meshsize or the time steps. Such stability estimates are then used to derive a priori error estimates with quasi-optimal convergence rates, where a suitable treatment of possible nonhomogeneous Dirichlet boundary conditions is pivotal to avoid loss of accuracy. Moreover, based on the properties of a postprocessed approximation, we derive a constant-free, reliable a posteriori error estimate in the $C^0([0, T]; L^2(\Omega))$ norm for the semidiscrete-in-time formulation. Several numerical experiments are presented to validate our theoretical findings.

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Sergio Gómez. 2025-01-20. A variational approach to the analysis of the continuous space-time FEM for the wave equation. https://arxiv.org/abs/2501.11494

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