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

TL;DR

This paper develops a novel continuous/discontinuous Galerkin finite element method for solving fourth order elliptic PDEs on surfaces, providing rigorous error estimates and demonstrating convergence through numerical experiments.

Contribution

It extends surface finite element methods to higher order differential operators on surfaces, with new a priori error estimates for biharmonic problems.

Findings

Error estimates in energy and L2 norms are proven.

Numerical experiments confirm theoretical convergence rates.

First analysis of higher order surface differential operators in this context.

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.