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arXiv · 2509.08645

Quasi-optimal time-space discretizations for a class of nonlinear parabolic PDEs

Nina Beranek·Robin Smeets·Rob Stevenson
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Abstract

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.

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Nina Beranek, Robin Smeets, Rob Stevenson. 2025-09-10. Quasi-optimal time-space discretizations for a class of nonlinear parabolic PDEs. https://arxiv.org/abs/2509.08645

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