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

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.