Quasi-optimal time-space discretizations for a class of nonlinear parabolic PDEs
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| Format: | Preprint |
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2025
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| _version_ | 1866908773860245504 |
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| author | Beranek, Nina Smeets, Robin Stevenson, Rob |
| author_facet | Beranek, Nina Smeets, Robin Stevenson, Rob |
| contents | We consider parabolic evolution equations with Lipschitz continuous and strongly monotone spatial operators. By introducing an additional variable, we construct an equivalent system where the operator is a Lipschitz continuous mapping from a Hilbert space $Y \times X$ to its dual, with a Lipschitz continuous inverse. Resulting Galerkin discretizations can be solved with an inexact Uzawa type algorithm. Quasi-optimality of the Galerkin approximations is guaranteed under an inf-sup condition on the selected `test' and `trial' subspaces of $Y$ and $X$. To circumvent the restriction imposed by this inf-sup condition, an a posteriori condition for quasi-optimality is developed that is shown to be satisfied whenever the test space is sufficiently large. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2509_08645 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Quasi-optimal time-space discretizations for a class of nonlinear parabolic PDEs Beranek, Nina Smeets, Robin Stevenson, Rob Numerical Analysis 35A15, 35B35, 35K90, 65M12, 65M15, 65M60 We consider parabolic evolution equations with Lipschitz continuous and strongly monotone spatial operators. By introducing an additional variable, we construct an equivalent system where the operator is a Lipschitz continuous mapping from a Hilbert space $Y \times X$ to its dual, with a Lipschitz continuous inverse. Resulting Galerkin discretizations can be solved with an inexact Uzawa type algorithm. Quasi-optimality of the Galerkin approximations is guaranteed under an inf-sup condition on the selected `test' and `trial' subspaces of $Y$ and $X$. To circumvent the restriction imposed by this inf-sup condition, an a posteriori condition for quasi-optimality is developed that is shown to be satisfied whenever the test space is sufficiently large. |
| title | Quasi-optimal time-space discretizations for a class of nonlinear parabolic PDEs |
| topic | Numerical Analysis 35A15, 35B35, 35K90, 65M12, 65M15, 65M60 |
| url | https://arxiv.org/abs/2509.08645 |