Adaptive discontinuous Galerkin methods on surfaces

2014/02/10 by Andreas Dedner, Dedner, Andreas, Pravin Madhavan +1
Engineering · Computer Science · #Advanced Numerical Methods in Computational Mathematics #Advanced Mathematical Modeling in Engineering #Numerical methods in engineering

paper · pdf · doi:10.48550/arxiv.1402.2117

Abstract

We present a dual weighted residual-based a posteriori error estimate for a discontinuous Galerkin (DG) approximation of a linear second-order elliptic problem on compact smooth connected and oriented surfaces in ℝ³ which are implicitly represented as level sets of a smooth function. 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.

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