An accurate and robust Eulerian finite element method for partial differential equations on evolving surfaces

In this paper we present a new Eulerian finite element method for the discretization of scalar partial differential equations on evolving surfaces. In this method we use the restriction of standard space-time finite element spaces on a fixed bulk mesh to the space-time surface. The structure of the method is such that it naturally fits to a level set representation of the evolving surface. The higher order version of the method is based on a space-time variant of a mesh deformation that has been developed in the literature for stationary surfaces. The discretization method that we present is of (optimal) higher order accuracy for smoothly varying surfaces with sufficiently smooth solutions. Without any modifications the method can be used for the discretization of problems with topological singularities. A numerical study demonstrates both the higher order accuracy for smooth cases and the robustness with respect to topological singularities.

Automated summary

In this paper, a new Eulerian finite element method is proposed for discretization of scalar partial differential equations on evolving surfaces. The method utilizes the restriction of standard space-time finite element spaces on a fixed bulk mesh to the space-time surface and is suitable for a level set representation of the evolving surface. The higher order version of the method is based on a space-time variant of a mesh deformation that is developed for stationary surfaces in the literature. The presented discretization method achieves (optimal) higher order accuracy for smoothly varying surfaces with sufficiently smooth solutions. It can also be used for problems with topological singularities without any modifications, as demonstrated in a numerical study.