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 R 3 \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 L^2 norm. This can be seen as an extension of the formalism and method originally used by Dziuk (1988) 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 Γ h \Gamma _h of an implicitly defined surface Γ \Gamma we employ continuous piecewise quadratic finite elements to approximate solutions to the biharmonic equation on Γ \Gamma . Numerical examples on the sphere and on the torus confirm the convergence rate implied by our estimates.