A HIGHER ORDER FINITE ELEMENT METHOD FOR PARTIAL DIFFERENTIAL EQUATIONS ON SURFACES

A new higher order finite element method for elliptic partial differential equations on a stationary smooth surface Gamma is introduced and analyzed. We assume that Gamma is characterized as the zero level of a level set function phi and only a finite element approximation phi(h) (of degree k >= 1) of phi is known. For the discretization of the partial differential equation, finite elements (of degree m >= 1) on a piecewise linear approximation of Gamma are used. The discretization is lifted to Gamma(h), which denotes the zero level of phi(h), using a quasi-orthogonal coordinate system that is constructed by applying a gradient recovery technique to phi(h). A complete discretization error analysis is presented in which the error is split into a geometric error, a quadrature error, and a finite element approximation error. The main result is a H-1(Gamma)-error bound of the form c(h(m)+h(k+1)). Results of numerical experiments illustrate the higher order convergence of this method.