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-infinity(0, T; L-2(Omega)) and L-2(0, T; H-1(Omega)) 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.