Space-time approximation of parabolic systems with variable growth

TL;DR

This paper establishes optimal convergence rates for finite element space-time approximations of parabolic systems with variable growth exponents, under certain regularity conditions on the exponent function.

Contribution

It provides the first rigorous analysis of error rates for parabolic systems with variable growth exponents in a space-time finite element framework.

Findings

Optimal convergence rate for gradient error in quasi norm

Error bounds hold under Hölder continuity of p(t,x)

Applicable to systems with variable growth conditions

Abstract

We study a parabolic system with $p(t,x)$-structure under Dirichlet boundary conditions. In particular, we deduce the optimal convergence rate for the error of the gradient of a finite element based space-time approximation. The error is measured in the quasi norm and the result holds if the exponent $p(t,x)$ is $(\alpha_t, \alpha_x)$-H\"{o}lder continuous.