Optimal Error Estimates for Fully Discrete Galerkin Approximations of Semilinear Parabolic Equations

TL;DR

This paper establishes uniform boundedness and optimal error estimates for fully discrete Galerkin methods applied to semilinear parabolic equations with broad nonlinearities, enhancing numerical analysis accuracy.

Contribution

It proves the uniform boundedness of the discrete solution for a wide class of nonlinearities, enabling optimal error estimates in various norms.

Findings

Proved uniform boundedness of the discrete solution.

Derived optimal error estimates in multiple norms.

Applicable to a broad class of nonlinearities without growth restrictions.

Abstract

We consider a semilinear parabolic equation with a large class of nonlinearities without any growth conditions. We discretize the problem with a discontinuous Galerkin scheme dG(0) in time (which is a variant of the implicit Euler scheme) and with conforming finite elements in space. The main contribution of this paper is the proof of the uniform boundedness of the discrete solution. This allows us to obtain optimal error estimates with respect to various norms.