Abstract. In this paper a new finite element approach for the discretization of elliptic partial differential equations on surfaces is treated. The main idea is to use finite element spaces that are induced by triangulations of an "outer" domain to discretize the partial differential equation on the surface. The method is particularly suitable for problems in which there is a coupling with a flow problem in an outer domain that contains the surface. We give an analysis that shows that the method has optimal order of convergence both in the H 1 -and in the L 2 -norm. Results of numerical experiments are included that confirm this optimality. 1. Introduction. Moving hypersurfaces and interfaces appear in many physical processes, for example, in multiphase flows and flows with free surfaces. Certain mathematical models involve elliptic partial differential equations posed on such surfaces. This happens, for example, in multiphase fluids if one takes so-called surface active agents (surfactants) into account. These surfactants induce tangential surface tension forces and thus cause Marangoni phenomena [9,10]. Numerical simulations play an important role in a better understanding and prediction of processes involving this or other surface phenomena. In mathematical models surface equations are often coupled with other equations that are formulated in a (fixed) domain which contains the surface. In such a setting a common approach is to use a splitting scheme that allows to solve at each time step a sequence of simpler (decoupled) equations. In doing so one has to solve numerically at each time step an elliptic type of equation on a surface. The surface may vary from one time step to another and usually only some discrete approximation of the surface is available. A well-known finite element method for solving elliptic equations on surfaces, initiated by the paper [5], consists of approximating the surface by a piecewise polygonal surface and using a finite element space on a triangulation of this discrete surface, cf. [3,9]. If the surface is changing in time, then this approach leads to time-dependent triangulations and time-dependent finite element spaces. Implementing this requires substantial data handling and programming effort. Another approach has recently been introduced in [2]. The method in that paper applies to cases in which the surface is given implicitly by some level set function, and the key idea is to solve the partial differential equation on a narrow band around the surface. Unfitted finite element spaces on this narrow band are used for discretization.

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 H 1 (Γ)-norm error bounds. A second topic of this paper is a unified analysis of several stabilization methods for trace finite element methods. Only a 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.

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Three new Eulerian finite element methods for parabolic PDEs on a moving surface Γ(t) are presented and compared in numerical experiments. These are space-time Galerkin methods, which are derived from a weak formulation in space and time. The trial-and test-spaces contain the traces on the space-time manifold of an outer prismatic finite element space. The numerical experiments show that two of the methods converge with second order with respect to both the time step size and the spatial mesh width. Introduction.In this paper, three new Eulerian finite element Galerkin methods for second order parabolic partial differential equations (PDEs) on an evolving smooth hypersurface Γ(t) ⊂ R 3 , t > 0, are proposed. The idea from [12] to use traces of "outer" finite element spaces is generalized to this setting in a space-time approach. Numerical experiments suggest that two of the methods converge with a rate that is second order in the spatial mesh width h x and time step size h t if the error is measured in an anisotropic Sobolev norm on the space-time manifold defined below. To our knowledge, these are the first Eulerian finite element methods which are second order accurate in space and time for parabolic PDEs on evolving surfaces.An application which involves such a PDE is two-phase flow with a surface active agent (surfactant). The surface concentration u of the surfactant on the phase interface Γ(t) determines the properties of Γ(t), for example, the surface tension. The PDE, which describes the evolution of u, is introduced in section 1.1. The decision to develop Eulerian methods for PDEs on a moving surface is partly based on the application to two-phase flow; for example, the implementation of the finite element spaces can easily be reused. This will be explained further in section 11. Additional details on two-phase flow can be found in [8].