Higher-Order Finite Element Methods and Pointwise Error Estimates for Elliptic Problems on Surfaces

Abstract

We define higher-order analogues to the piecewise linear surface finite element method studied in [G. Dziuk, “Finite elements for the Beltrami operator on arbitrary surfaces,” in Partial Differential Equations and Calculus of Variations, Springer-Verlag, Berlin, 1988, pp. 142-155] and prove error estimates in both pointwise and $L_2$-based norms. Using the Laplace-Beltrami problem on an implicitly defined surface $\Gamma$ as a model PDE, we define Lagrange finite element methods of arbitrary degree on polynomial approximations to $\Gamma$ which likewise are of arbitrary degree. Then we prove a priori error estimates in the $L_2$, $H^1$, and corresponding pointwise norms that demonstrate the interaction between the “PDE error” that arises from employing a finite-dimensional finite element space and the “geometric error” that results from approximating $\Gamma$. We also consider parametric finite element approximations that are defined on $\Gamma$ and thus induce no geometric error. Computational examples confirm the sharpness of our error estimates.