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An ALE ESFEM for solving PDEs on evolving surfaces. (English) Zbl 1259.65147

Summary: Numerical methods for approximating the solution of partial differential equations (PDEs) on evolving hypersurfaces using surface finite elements on evolving triangulated surfaces are presented. In the arbitrary Eulerian Lagrangian evolving surface finite element method (ALE ESFEM), the vertices of the triangles evolve with a velocity which is normal to the hypersurface whilst having a tangential velocity which is arbitrary. This is in contrast to the original evolving surface finite element method in which the nodes move with a material velocity. Numerical experiments are presented which illustrate the value of choosing the arbitrary tangential velocity to improve mesh quality. Simulations of two applications arising in material science and biology are presented which couple the evolution of the surface to the solution of the surface partial differential equation.

MSC:

65M60 Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs
65M15 Error bounds for initial value and initial-boundary value problems involving PDEs
35R01 PDEs on manifolds
35R37 Moving boundary problems for PDEs
35K20 Initial-boundary value problems for second-order parabolic equations
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