An interior-penalty discontinuous Galerkin (dG) method for an elliptic interface problem involving, possibly, curved interfaces, with flux-balancing interface conditions, e.g., modelling mass transfer of solutes through semipermeable 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.

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 semipermeable membranes and other filtering processes. The adaptive algorithm is based on a residualtype 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 Neumannand/or Robin-type boundary conditions, posed on general curved boundaries, also follows. Numerical experiments are also presented.