A note on exponential Rosenbrock-Euler method for the finite element discretization of a semilinear parabolic partial differential equation

TL;DR

This paper presents a rigorous convergence analysis of the exponential Rosenbrock-Euler method combined with finite element discretization for semilinear parabolic PDEs, achieving optimal convergence orders under standard conditions.

Contribution

It provides the first convergence proof in both space and time for this method under only Lipschitz conditions, unlike previous restrictive assumptions.

Findings

Achieves optimal convergence orders in space and time.

Works for both smooth and nonsmooth initial conditions.

Provides rigorous theoretical convergence proof.

Abstract

In this paper we consider the numerical approximation of a general second order semi-linear parabolic partial differential equation. Equations of this type arise in many contexts, such as transport in porous media. Using finite element method for space discretization and the exponential Rosenbrock-Euler method for time discretization, we provide a rigorous convergence proof in space and time under only the standard Lipschitz condition of the nonlinear part for both smooth and nonsmooth initial solution. This is in contrast to very restrictive assumptions made in the literature, where the authors have considered only approximation in time so far in their convergence proofs. The optimal orders of convergence in space and in time are achieved for smooth and nonsmooth initial solution.