Non-degenerate Eulerian finite element method for solving PDEs on surfaces

Abstract

The paper studies a method for solving elliptic partial differential equations posed on hypersurfaces in ℝN, N = 2; 3. The method builds upon the formulation introduced in [7], where a surface equation is extended to a neighbourhood of the surface. The resulting degenerate PDE is then solved in one dimension higher, but can be solved on a mesh that is unaligned to the surface. We introduce another extended formulation, which leads to uniformly elliptic (non-degenerate) equations in a bulk domain containing the surface.We apply a finite element method to solve this extended PDE and prove the convergence of the finite element solutions restricted to the surface to the solution of the original surface problem. Several numerical examples illustrate the properties of the method.