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.