A continuous/discontinuous Galerkin method and a priori error estimates for the biharmonic problem on surfaces

Abstract

We present a continuous/discontinuous Galerkin method for approximating solutions to a fourth order elliptic PDE on a surface embedded in ${\mathbb{R}}^3$. A priori error estimates, taking both the approximation of the surface and the approximation of surface differential operators into account, are proven in a discrete energy norm and in $L^2$-norm. This can be seen as an extension of the formalism and method originally used by Dziuk [14] for approximating solutions to the Laplace-Beltrami problem, and within this setting this is the first analysis of a surface finite element method formulated using higher order surface differential operators. Using a polygonal approximation ${\Gamma }_h$ of an implicitly defined surface $\Gamma$ we employ continuous piecewise quadratic finite elements to approximate solutions to the biharmonic equation on $\Gamma$. Numerical examples on the sphere and on the torus confirm the convergence rate implied by our estimates.

41 pages, LaTeX

References in corpus (2)

• Analysis of the discontinuous Galerkin method for elliptic problems on surfaces
• A Stabilized Cut Finite Element Method for the Darcy Problem on Surfaces

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