A priori and a posteriori error estimates of a $\mathcal C^0$-in-time method for the wave equation in second order formulation
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arXiv
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| Natura: | Preprint |
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2024
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| _version_ | 1866914526629199872 |
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| author | Dong, Zhaonan Mascotto, Lorenzo Wang, Zuodong |
| author_facet | Dong, Zhaonan Mascotto, Lorenzo Wang, Zuodong |
| contents | We establish fully-discrete a priori and semi-discrete in time a posteriori error estimates for a discontinuous-continuous Galerkin discretization of the wave equation in second order formulation; the resulting method is a Petrov-Galerkin scheme based on piecewise polynomial test functions and continuous piecewise polynomial trial functions in time, respectively. Crucial tools in the a priori analysis for the fully-discrete formulation are the design of suitable projection and interpolation operators extending those used in the parabolic setting, and stability estimates based on a nonstandard choice of the test function; a priori estimates are shown, which are measured in $L^\infty$-type norms in time. For the semi-discrete in time formulation, we exhibit reliable a posteriori error estimates for the error measured in the $L^\infty(L^2)$ norm with fully explicit constants; to this aim, we design a reconstruction operator into $\mathcal C^1$ piecewise polynomials over the time grid with optimal approximation properties in terms of the polynomial degree distribution and the time steps. Numerical examples illustrate the theoretical findings. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_03264 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A priori and a posteriori error estimates of a $\mathcal C^0$-in-time method for the wave equation in second order formulation Dong, Zhaonan Mascotto, Lorenzo Wang, Zuodong Numerical Analysis 65M50, 65M60, 65J10 We establish fully-discrete a priori and semi-discrete in time a posteriori error estimates for a discontinuous-continuous Galerkin discretization of the wave equation in second order formulation; the resulting method is a Petrov-Galerkin scheme based on piecewise polynomial test functions and continuous piecewise polynomial trial functions in time, respectively. Crucial tools in the a priori analysis for the fully-discrete formulation are the design of suitable projection and interpolation operators extending those used in the parabolic setting, and stability estimates based on a nonstandard choice of the test function; a priori estimates are shown, which are measured in $L^\infty$-type norms in time. For the semi-discrete in time formulation, we exhibit reliable a posteriori error estimates for the error measured in the $L^\infty(L^2)$ norm with fully explicit constants; to this aim, we design a reconstruction operator into $\mathcal C^1$ piecewise polynomials over the time grid with optimal approximation properties in terms of the polynomial degree distribution and the time steps. Numerical examples illustrate the theoretical findings. |
| title | A priori and a posteriori error estimates of a $\mathcal C^0$-in-time method for the wave equation in second order formulation |
| topic | Numerical Analysis 65M50, 65M60, 65J10 |
| url | https://arxiv.org/abs/2411.03264 |