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

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 as a model PDE, we define Lagrange finite element methods of arbitrary degree on polynomial approximations to 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 . We also consider parametric finite element approximations that are defined on and thus induce no geometric error. Computational examples confirm the sharpness of our error estimates. 2009 Society for Indust rial and Applied Mathematics.

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