A variational approach to the analysis of the continuous space-time FEM for the wave equation
| Title: | A variational approach to the analysis of the continuous space-time FEM for the wave equation |
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| Authors: | Gómez, Sergio |
| Publication Year: | 2025 |
| Collection: | Computer Science Mathematics |
| Subject Terms: | Mathematics - Numerical Analysis, 65M60, 65M12, 35L04 |
| More Details: | 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. |
| Document Type: | Working Paper |
| Access URL: | http://arxiv.org/abs/2501.11494 |
| Accession Number: | edsarx.2501.11494 |
| Database: | arXiv |
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