Higher-order exponential Runge-Kutta Galerkin finite element method for semilinear parabolic problems with nonsmooth data

Abstract

We develop a rigorous numerical analysis framework for a class of semilinear parabolic problems with nonsmooth initial data. We employ a linear Galerkin finite element method for spatial discretization coupled with a high-order explicit exponential Runge-Kutta (EERK) temporal integration scheme. In contrast to conventional smooth error analysis, the nonsmooth case lacks a priori estimates for the higher-order derivatives of both the nonlinear term and the exact solution. By combining analytic semigroup techniques with fractional power space theory, we establish rigorous bounds for these derivatives. Finally, our analysis proves that the $p$th-order EERK method achieves a convergence rate of $\min(1 + γ/2 + ρ_1(γ)/2,\:p)$, where $γ$ characterizes the initial data regularity and $ρ_1(γ)$ quantifies the boundedness of the nonlinearity's first Fréchet derivative. Numerical experiments confirm the sharpness of these estimates.