Numerical Framework for Pattern-Forming Models on Evolving-in-Time Surfaces

In this contribution we describe a numerical framework for a system of coupledreaction-diffusion equations  on an evolving-in-time hypersurface$\Gamma$. Numerical tests are empolyed for Turing-type instability on stationary and evolving surfaces. The proposed framework combines the level set methodology forthe implicit description of the time dependent Г , the Eulerianfinite element formulation for the numerical treatment of partialdifferential equations, and the flux-corrected transport scheme for thenumerical stabilization of arising adjective, resp., convective terms.Major advantages of this scheme are that it avoids numerical calculation ofcurvature, allows coupling of surface-defined partialdifferential equations with domain-defined partialdifferential equations through the level set bulk and preserves the positivity of the solution throughthe algebraic flux correction. Corresponding numerical tests demonstratethe ability of the scheme to deliver  highly accurate solutions with a reasonably good convergence behavior.