Preserving invariance properties of reaction–diffusion systems on stationary surfaces

Abstract

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