Analysis of a High-Order Trace Finite Element Method for PDEs on Level Set Surfaces

We present a new high order finite element method for the discretization of partial differential equations on stationary smooth surfaces which are implicitly described as the zero level of a level set function. The discretization is based on a trace finite element technique. The higher discretization accuracy is obtained by using an isoparametric mapping of the volume mesh, based on the level set function, as introduced in [C. Lehrenfeld, High order unfitted finite element methods on level set domains using isoparametric mappings, Comp. Meth. Appl. Mech. Engrg. 2016]. The resulting trace finite element method is easy to implement. We present an error analysis of this method and derive optimal order H1(Γ)-norm error bounds. A second main topic of this paper is a unified analysis of several stabilization methods for trace finite element methods. Three methods known from the literature and one new method are analyzed in a general framework. Only the new stabilization method, which is based on adding an anisotropic diffusion in the volume mesh, is able to control the condition number of the stiffness matrix also for the case of higher order discretizations. Results of numerical experiments are included which confirm the theoretical findings on optimal order discretization errors and uniformly bounded condition numbers.