Implicit-Explicit Timestepping with Finite Element Approximation of Reaction-Diffusion Systems on Evolving Domains

We present and analyze an implicit–explicit timestepping procedure with finite element spatial approximation for 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 L∞(0, T ; L2(Ω)) and L2(0, T ; H (Ω)) 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.