EFFICIENT EXPONENTIAL INTEGRATOR FINITE ELEMENT METHOD FOR SEMILINEAR PARABOLIC EQUATIONS

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 semidiscrete system to produce a fully discrete numerical solution. Under certain regularity assumptions, error estimates measured in H1norm are successfully derived for the proposed schemes with one and two Runge--Kutta 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 the 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.