Convergence of an adaptive discontinuous Galerkin method for elliptic interface problems

We prove a basic error contraction result of an adaptive discontinuous Galerkin method for an elliptic interface problem. The interface conditions considered model mass transfer of solutes through semi-permeable membranes and other filtering processes. The adaptive algorithm is based on a residual-type a posteriori error estimator, with a bulk refinement criterion. The a posteriori error bound is derived under the assumption that the triangulation is aligned with the interfaces although, crucially, extremely general curved element shapes are also allowed, resolving the interface geometry exactly. As a corollary, convergence of the adaptive discontinuous Galerkin method for non-essential Neumann-and/or Robin-type boundary conditions, posed on general curved boundaries, also follows. Numerical experiments are also presented. Keywords: Discontinuous Galerkin method, interface problem, a posteriori error bound, adaptivity, convergence analysis, a posteriori error analysis on curved domains