A least-squares space-time approach for parabolic equations
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arXiv
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| Main Authors: | , , |
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| Format: | Preprint |
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2023
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| _version_ | 1866913998746681344 |
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| author | Hinze, Michael Kahle, Christian Stahl, Michael |
| author_facet | Hinze, Michael Kahle, Christian Stahl, Michael |
| contents | We propose a least squares formulation for abstract parabolic equations in the natural $L^2(0,T;V^\star)\times H$ norm which only relies on natural regularity assumptions on the data of the problem. The resulting bilinear form then is symmetric, coercive and continuous. We provide two space-time Galerkin frameworks for the numerical approximation. The first one uses a conformal discretization of the underlying bilinear system and relies on the fact that the $V^*-$norm of basis functions can be evaluated exactly. The second approach is nonconforming an replaces the evaluation of the $V^*-$norm by a discrete pendant. We prove convergence for both approaches and illustrate our analytical findings by selected numerical experiments. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2305_03402 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | A least-squares space-time approach for parabolic equations Hinze, Michael Kahle, Christian Stahl, Michael Numerical Analysis We propose a least squares formulation for abstract parabolic equations in the natural $L^2(0,T;V^\star)\times H$ norm which only relies on natural regularity assumptions on the data of the problem. The resulting bilinear form then is symmetric, coercive and continuous. We provide two space-time Galerkin frameworks for the numerical approximation. The first one uses a conformal discretization of the underlying bilinear system and relies on the fact that the $V^*-$norm of basis functions can be evaluated exactly. The second approach is nonconforming an replaces the evaluation of the $V^*-$norm by a discrete pendant. We prove convergence for both approaches and illustrate our analytical findings by selected numerical experiments. |
| title | A least-squares space-time approach for parabolic equations |
| topic | Numerical Analysis |
| url | https://arxiv.org/abs/2305.03402 |