Milan Journal of Mathematics

, Volume 80, Issue 2, pp 469–501

# An ALE ESFEM for Solving PDEs on Evolving Surfaces

Article

## Abstract

Numerical methods for approximating the solution of partial differential equations on evolving hypersurfaces using surface finite elements on evolving triangulated surfaces are presented. In the 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.

### Mathematics Subject Classification (2010)

Primary 65M60 65M15 Secondary 35K99 35R01 35R37 76R99

### Keywords

Surface finite elements ALE advection diffusion equation interface motion

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