Implicit-explicit timestepping with finite element approximation of reaction-diffusion systems 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.

References in corpus (2)

• Large mass self-similar solutions of the parabolic-parabolic Keller--Segel model of chemotaxis
• Global existence for semilinear reaction-diffusion systems on evolving domains

Cited by in corpus (4)

• Stability analysis and simulations of coupled bulk-surface reaction-diffusion systems
• Fully implicit time-stepping schemes and non-linear solvers for systems of reaction-diffusion equations
• Lumped finite element method for reaction-diffusion systems on compact surfaces
• Parallel two-scale finite element implementation of a system with varying microstructures