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We present a new high order finite element method for the discretization ofpartial differential equations on stationary smooth surfaces which areimplicitly described as the zero level of a level set function. Thediscretization is based on a trace finite element technique. The higherdiscretization accuracy is obtained by using an isoparametric mapping of thevolume mesh, based on the level set function, as introduced in [C. Lehrenfeld,\emph{High order unfitted finite element methods on level set domains usingisoparametric mappings}, Comp. Meth. Appl. Mech. Engrg. 2016]. The resultingtrace finite element method is easy to implement. We present an error analysisof this method and derive optimal order $H^1(\Gamma)$-norm error bounds. Asecond topic of this paper is a unified analysis of several stabilizationmethods for trace finite element methods. Only a stabilization method which isbased on adding an anisotropic diffusion in the volume mesh is able to controlthe condition number of the stiffness matrix also for the case of higher orderdiscretizations. Results of numerical experiments are included which confirmthe theoretical findings on optimal order discretization errors and uniformlybounded condition numbers.

Analysis of a high order Trace Finite Element Method for PDEs on level set surfaces