403 | IGPM403.pdf July 2014 |

TITLE | A Higher Order Finite Element Method for Partial Diﬀerential Equations on Surfaces |

AUTHORS | Jörg Grande, Arnold Reusken |

ABSTRACT | 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 |

DOI | 10.1137/14097820X |

PUBLICATION | SIAM Journal Numerical Analysis 54(1), 388–414. (27 pages) |