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arXiv · 1111.5052

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

Omar Lakkis·Anotida Madzvamuse·Chandrasekhar Venkataraman
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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.

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Omar Lakkis, Anotida Madzvamuse, Chandrasekhar Venkataraman. 2011-11-21. Implicit-explicit timestepping with finite element approximation of reaction-diffusion systems on evolving domains. https://doi.org/10.1137/120880112

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