|Preprint-No.:||< 403 >||Published in:||July 2014||PDF-File:||IGPM403.pdf|
|Title:||A Higher Order Finite Element Method for Partial Diﬀerential Equations on Surfaces|
|Authors:||Jörg Grande, Arnold Reusken|
A new higher order ﬁnite element method for elliptic partial differential equations on a stationary smooth surface Γ is introduced and analyzed. We assume 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.
|Keywords:||Laplace Beltrami equation, surface ﬁnite element method, high order, gradient recovery, error analysis|
|Publication:||SIAM J. Numer. Anal., 54(1), 388–414. (27 pages)|