Higher-order exponential Runge-Kutta Galerkin finite element method for semilinear parabolic problems with nonsmooth data
This work provides rigorous convergence bounds for high-order exponential integrators applied to semilinear parabolic PDEs with nonsmooth data, a setting where standard smooth error analysis fails.
The authors develop a rigorous numerical analysis for semilinear parabolic problems with nonsmooth initial data, proving that a high-order explicit exponential Runge-Kutta method achieves a convergence rate of min(1 + γ/2 + ρ₁(γ)/2, p). Numerical experiments confirm the sharpness of these estimates.
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.