A variational approach to the analysis of the continuous space-time FEM for the wave equation
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866915395785457664 |
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| author | Gómez, Sergio |
| author_facet | Gómez, Sergio |
| contents | 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 $C^0([0, T]; L^2(Ω))$ norm for the semidiscrete-in-time formulation. Several numerical experiments are presented to validate our theoretical findings. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2501_11494 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A variational approach to the analysis of the continuous space-time FEM for the wave equation Gómez, Sergio Numerical Analysis 65M60, 65M12, 35L04 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 $C^0([0, T]; L^2(Ω))$ norm for the semidiscrete-in-time formulation. Several numerical experiments are presented to validate our theoretical findings. |
| title | A variational approach to the analysis of the continuous space-time FEM for the wave equation |
| topic | Numerical Analysis 65M60, 65M12, 35L04 |
| url | https://arxiv.org/abs/2501.11494 |