Analysis of the discontinuous Galerkin method for elliptic problems on surfaces

Details Analysis of the discontinuous Galerkin method for elliptic problems on surfaces Analysis of the discontinuous Galerkin method for elliptic problems on surfaces

2012-03-25

arxiv.org/abs/1203.5531v2

Mathematics

Numerical Analysis

20 pages, 4 figures

Scientific paper

We extend the discontinuous Galerkin (DG) framework to a linear second-order elliptic problem on a compact smooth connected and oriented surface. An interior penalty (IP) method is introduced on a discrete surface and we derive a-priori error estimates by relating the latter to the original surface via the lift introduced in Dziuk (1988). The estimates suggest that the geometric error terms arising from the surface discretisation do not affect the overall convergence rate of the IP method when using linear ansatz functions. This is then verified numerically for a number of test problems. An intricate issue is the approximation of the surface conormal required in the IP formulation, choices of which are investigated numerically. Furthermore, we present a generic implementation of test problems on surfaces.

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