Efficient Exponential Integrator Finite Element Method for Semilinear Parabolic Equations

TL;DR

This paper introduces an efficient exponential integrator finite element method for semilinear parabolic equations, combining finite element spatial discretization with explicit exponential Runge-Kutta time integration, achieving fast solutions and validated by numerical experiments.

Contribution

It develops a novel exponential integrator finite element scheme with diagonalizable matrices for faster computation and provides error estimates under regularity assumptions.

Findings

Error estimates in $H^1$-norm are derived.

The method achieves fast solution times via spectral decomposition.

Numerical experiments confirm theoretical accuracy and efficiency.

Abstract

In this paper, we propose an efficient exponential integrator finite element method for solving a class of semilinear parabolic equations in rectangular domains. The proposed method first performs the spatial discretization of the model equation using the finite element approximation with continuous multilinear rectangular basis functions, and then takes the explicit exponential Runge-Kutta approach for time integration of the resulting semi-discrete system to produce fully-discrete numerical solution. Under certain regularity assumptions, error estimates measured in $H^1$-norm are successfully derived for the proposed schemes with one and two RK stages. More remarkably, the mass and coefficient matrices of the proposed method can be simultaneously diagonalized with an orthogonal matrix, which provides a fast solution process based on tensor product spectral decomposition and fast Fourier transform. Various numerical experiments in two and three dimensions are also carried out to validate the theoretical results and demonstrate the excellent performance of the proposed method.