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

2025/01/20 by Gómez, Sergio · 2 citations
#35L04 #65M12 #65M60 #FOS: Mathematics #Numerical Analysis (math.NA)

paper · doi:10.48550/arxiv.2501.11494

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 C⁰([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 C⁰([0, T]; L²(Ω)) norm for the semidiscrete-in-time formulation. Several numerical experiments are presented to validate our theoretical findings.

Cited by

• Space-time finite element methods for nonlinear wave equations via elliptic regularisation
• Inf-sup stable space-time discretization of the wave equation based on a first-order-in-time variational formulation

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