Exponential Runge-Kutta Galerkin finite element method for a reaction-diffusion system with nonsmooth initial data

TL;DR

This paper develops a numerical method combining exponential Runge-Kutta and Galerkin finite elements to solve reaction-diffusion systems with nonsmooth initial data, providing error estimates that adapt to data regularity.

Contribution

It introduces a novel error analysis framework for reaction-diffusion equations with nonsmooth initial data using fractional Sobolev spaces and semigroup techniques.

Findings

Error estimates in L2 and H1 norms that depend on initial data smoothness

Convergence order adapts to the regularity of initial data

Numerical examples validate theoretical results

Abstract

This study presents a numerical analysis of the Field-Noyes reaction-diffusion model with nonsmooth initial data, employing a linear Galerkin finite element method for spatial discretization and a second-order exponential Runge-Kutta scheme for temporal integration. The initial data are assumed to reside in the fractional Sobolev space H^gamma with 0 < gamma < 2, where classical regularity conditions are violated, necessitating specialized error analysis. By integrating semigroup techniques and fractional Sobolev space theory, sharp fully discrete error estimates are derived in both L2 and H1 norms. This demonstrates that the convergence order adapts to the smoothness of initial data, a key advancement over traditional approaches that assume higher regularity. Numerical examples are provided to support the theoretical analysis.