Mathematics – Numerical Analysis
Scientific paper
2011-11-21
Mathematics
Numerical Analysis
Scientific paper
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 L{\infty}(0, T ; L2 ({\Omega})) and L2 (0, T ; H1 ({\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 finally report on numerical experiments for an application to pattern formation on evolving domains, where we observe novel complex pattern transitions in multicomponent systems and the emergence of patterns with different wavelengths under nonuniform evolution.
Lakkis Omar
Madzvamuse Anotida
Venkataraman Chandrasekhar
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