Quasi-Optimal Time-Space Discretizations for a Class of Nonlinear Parabolic PDEs

Abstract

SIAM Journal on Numerical Analysis, Volume 64, Issue 3, Page 804-827, June 2026.
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 [math] 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 [math] and [math]. 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.