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 Γ is introduced and analyzed. We assume that Γ is characterized as the zero level of a level set function φ and only a finite element approximation φₕ (of degree k ≥ 1) of φ is known. For the discretization of the partial differential equation, finite elements (of degree m≥ 1) on a piecewise linear approximation of Γ are used. The discretization is lifted to Γₕ, which denotes the zero level of φₕ, using a quasi-orthogonal coordinate system that is constructed by applying a gradient recovery technique to φₕ. 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 H¹(Γ)-error bound of the form c(hᵐ + hᵏ⁺¹). Results of numerical experiments illustrate the higher order convergence of this method.