Implicit-explicit timestepping with finite element approximation of reaction-diffusion systems on evolving domains

TL;DR

This paper develops and analyzes an implicit-explicit finite element method for reaction-diffusion systems on evolving biological domains, providing error estimates and stability results, with applications to pattern formation.

Contribution

It introduces a novel Lagrangian-based error analysis for reaction-diffusion equations on moving domains, including optimal error estimates and stability proofs.

Findings

Optimal error estimates in key norms.

Stability results applicable to Eulerian solutions.

Numerical demonstration of pattern formation on evolving domains.

Abstract

We present and analyse an implicit-explicit timestepping procedure with finite element spatial approximation for a semilinear reaction-diffusion systems on evolving domains arising from biological models, such as Schnakenberg's (1979). We employ a Lagrangian formulation of the model equations which permits the error analysis for parabolic equations on a fixed domain but introduces technical difficulties, foremost the space-time dependent conductivity and diffusion. We prove optimal-order error estimates in the $\Lp{\infty}(0,T;\Lp{2}(\W))$ and $\Lp{2}(0,T;\Hil{1}(\W))$ norms, and a pointwise stability result. We remark that these apply to Eulerian solutions. Details on the implementation of the Lagrangian and the Eulerian scheme are provided. We also report on a numerical experiment for an application to pattern formation on an evolving domain.