Adaptive discontinuous Galerkin methods on surfaces

作者：Dedner Andreas; Madhavan Pravin^*

来源：Numerische Mathematik, 2016, 132(2): 369-398.

DOI：10.1007/s00211-015-0719-4

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 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 "geometric 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.

• 出版日期2016-2