Adaptive discontinuous Galerkin methods on surfaces

Abstract

We present a dual weighted residual-based a posteriori error estimate for a discontinuous Galerkin approximation of a surface partial differential equation. We restrict our analysis to a linear second-order elliptic problem posed on hypersurfaces in $$\mathbb {R}^{3}$$R3 which are implicitly represented as level sets of smooth functions. We show that the error in the energy norm may be split into a residual part and a higher order part. Upper and lower bounds for the resulting a posteriori error estimator are proven and we consider a number of challenging test problems to demonstrate the reliability and efficiency of the estimator. We also present a novel geometric driven refinement strategy for PDEs on surfaces which considerably improves the performance of the method on complex surfaces.