A posteriori error analysis and adaptivity of a space-time finite element method for the wave equation in second order formulation
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| Main Authors: | , , , |
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| Format: | Preprint |
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2025
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| _version_ | 1866915951275933696 |
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| author | Dong, Zhaonan Georgoulis, Emmanuil H. Mascotto, Lorenzo Wang, Zuodong |
| author_facet | Dong, Zhaonan Georgoulis, Emmanuil H. Mascotto, Lorenzo Wang, Zuodong |
| contents | We establish rigorous \emph{a posteriori} error bounds for a space-time finite element method of arbitrary order discretising linear wave problems in second order formulation. The method combines standard finite elements in space and continuous piecewise polynomials in time with an upwind discontinuous Galerkin-type approximation for the second temporal derivative. The proposed scheme accepts dynamic mesh modification, as required by space-time adaptive algorithms, resulting in a discontinuous temporal discretisation when mesh changes occur. We prove \emph{a posteriori} error bounds in the $L^\infty(L^2)$-norm, using carefully designed temporal and spatial reconstructions; explicit control on the constants (including the spatial and temporal orders of the method) in those error bounds is shown. The convergence behaviour of an error estimator is verified numerically, also taking into account the effect of the mesh change. A space-time adaptive algorithm is proposed and tested numerically. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2509_08537 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A posteriori error analysis and adaptivity of a space-time finite element method for the wave equation in second order formulation Dong, Zhaonan Georgoulis, Emmanuil H. Mascotto, Lorenzo Wang, Zuodong Numerical Analysis 65N30, 65M50, 65M60, 65J10 We establish rigorous \emph{a posteriori} error bounds for a space-time finite element method of arbitrary order discretising linear wave problems in second order formulation. The method combines standard finite elements in space and continuous piecewise polynomials in time with an upwind discontinuous Galerkin-type approximation for the second temporal derivative. The proposed scheme accepts dynamic mesh modification, as required by space-time adaptive algorithms, resulting in a discontinuous temporal discretisation when mesh changes occur. We prove \emph{a posteriori} error bounds in the $L^\infty(L^2)$-norm, using carefully designed temporal and spatial reconstructions; explicit control on the constants (including the spatial and temporal orders of the method) in those error bounds is shown. The convergence behaviour of an error estimator is verified numerically, also taking into account the effect of the mesh change. A space-time adaptive algorithm is proposed and tested numerically. |
| title | A posteriori error analysis and adaptivity of a space-time finite element method for the wave equation in second order formulation |
| topic | Numerical Analysis 65N30, 65M50, 65M60, 65J10 |
| url | https://arxiv.org/abs/2509.08537 |