An Adaptive Nonsymmetric Finite Volume and Boundary Element Coupling Method for a Fluid Mechanics Interface Problem

Abstract

We consider an interface problem often arising in transport problems: a coupled system of partial differential equations with one (elliptic) transport equation on a bounded domain and one equation (in this case the Laplace problem) on the complement, an unbounded domain. Based on the nonsymmetric coupling of the finite volume method and boundary element method of [C. Erath, G. Of, and F.-J. Sayas, Numer. Math., 135 (2017), pp. 895--922] we introduce a semirobust residual error estimator. The upper bound of the error in an energy (semi)norm is robust against variation of the model data. The lower bound, however, additionally depends on the Péclet number and is therefore only semirobust. In several examples we use the local contributions of the a posteriori error estimator to steer an adaptive mesh-refining algorithm. The adaptive finite volume method--boundary element method coupling turns out to be an efficient method especially for solving problems from fluid mechanics, mainly because of the local flux conservation and the stable approximation of convection dominated problems.