Adaptive discontinuous Galerkin methods for elliptic interface problems

Cangiani, Andrea; Georgoulis, Emmanuil; Sabawi, Younis

doi:10.1090/mcom/3322

Adaptive discontinuous Galerkin methods for elliptic interface problems
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by Andrea Cangiani, Emmanuil H. Georgoulis and Younis A. Sabawi PDF

Math. Comp. 87 (2018), 2675-2707 Request permission

Abstract:

An interior-penalty discontinuous Galerkin (dG) method for an \penalty-10000 elliptic interface problem involving, possibly, curved interfaces, with flux-\penalty-10000 balancing interface conditions, e.g., modelling mass transfer of solutes through semi-permeable membranes, is considered. The method allows for extremely general curved element shapes employed to resolve the interface geometry exactly. A residual-type a posteriori error estimator for this dG method is proposed and upper and lower bounds of the error in the respective dG-energy norm are proven. The a posteriori error bounds are subsequently used to prove a basic a priori convergence result. The theory presented is complemented by a series of numerical experiments. The presented approach applies immediately to the case of curved domains with non-essential boundary conditions, too.

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