Implicit-explicit timestepping with finite element approximation of reaction-diffusion systems on evolving domains
This work provides rigorous error analysis for a numerical method solving reaction-diffusion systems on evolving domains, which is relevant for biological pattern formation models.
The authors develop and analyze an implicit-explicit timestepping method with finite element spatial approximation for semilinear reaction-diffusion systems on evolving domains, proving optimal-order error estimates and stability. Numerical experiments on pattern formation demonstrate the method's applicability.
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.