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

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.

34 pages, 5 figures, 4 tables

References in corpus (3)

• Evolving surface finite element method for the Cahn-Hilliard equation
• Number of paths versus number of basis functions in American option pricing
• Maximum norm stability and error estimates for the evolving surface finite element method

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• Matrix-oriented FEM formulation for stationary and time-dependent PDEs on x-normal domains
• Discrete maximum principles for nonlinear elliptic finite element problems on Riemannian manifolds with boundary