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

TL;DR

This work develops a time-space variational framework for nonlinear parabolic PDEs with a Lipschitz continuous and strongly monotone spatial operator by introducing a secondary variable to form a $2\times2$ system whose operator is Lipschitz with a Lipschitz inverse. The authors prove quasi-optimality of Galerkin discretizations under a uniform inf-sup condition and present an a posteriori criterion to relax this requirement, enabling adaptive enrichment of the test space. An inexact Uzawa algorithm is analyzed for solving the resulting nonlinear Galerkin systems, with a posteriori error estimators to guide accuracy and enrichment. They further develop preconditioning strategies for tensor-product discretizations to achieve efficient, scalable linear-time applications of inverses, paving the way for fully adaptive double time-space refinement in nonlinear parabolic problems.

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.

Theorems & Definitions (42)

• example 2.1: Heat equation
• example 2.2: Quasi-linear parabolic PDE
• Theorem 4.1
• remark 4.2
• remark 4.3
• Lemma 4.4: 249.99
• proof
• Corollary 4.5
• Proposition 4.6
• proof