Lumped finite element method for reaction-diffusion systems on compact surfaces

TL;DR

This paper introduces a novel finite element method for reaction-diffusion systems on compact surfaces that preserves invariant regions and achieves optimal error bounds, supported by theoretical analysis and numerical experiments.

Contribution

It develops a lumped finite element method that maintains invariant regions for reaction-diffusion systems on surfaces, with proven stability and convergence properties.

Findings

Preservation of invariant rectangles under discretization.

Optimal quadratic convergence in space and linear in time.

Numerical examples demonstrating the importance of lumping for stability.

Abstract

We propose and analyse a novel surface finite element method that preserves the invariant regions of systems of semilinear parabolic equations on closed compact surfaces in $\mathbb{R}^3$ under discretisation. We also provide a fully-discrete scheme by applying the implicit-explicit (IMEX) Euler method in time. We prove the preservation of the invariant rectangles of the continuous problem under spatial and full discretizations. For scalar equations, these results reduce to the well-known discrete maximum principle. Furthermore, we prove optimal error bounds for the semi- and fully-discrete methods, that is the convergence rates are quadratic in the meshsize and linear in the timestep. Numerical experiments are provided to support the theoretical findings. In particular we provide examples in which, in the absence of lumping, the numerical solution violates the invariant region leading to blow-up due to the nature of the kinetics.