A note on exponential Rosenbrock-Euler method for the finite element discretization of a semilinear parabolic partial differential equation

In this paper, we consider the numerical approximation of a general second order semilinear parabolic partial differential equation. Equations of this type arise in many contexts, such as transport in porous media. Using finite element method for space discretization and the exponential Rosenbrock-Euler method for time discretization, we provide a convergence proof in space and time under only the standard Lipschitz condition of the nonlinear part, for both smooth and nonsmooth initial solutions. This is in contrast to restrictive assumptions made in the literature, where the authors have considered only approximation in time so far in their convergence proofs. The main result reveals how the convergence orders in both space and time depend heavily on the regularity of the initial data. In particular, the method achieves optimal convergence order Oh2+Δt2 when the initial data belongs to the domain of the linear operator. Numerical simulations to sustain our theoretical result are provided.