A posteriori error estimates and adaptive mesh refinement for the coupling of the finite volume method and the boundary element method

摘要

We consider the coupling of the finite volume method and the boundary element method of Erath [SIAM J. Numer. Anal., 50 (2012), pp. 574-594] in two and three dimensions. This method can be used, for example, to approximate a solution of the transport of a concentration in a fluid, where no boundary conditions are available. We derive residual-based a posteriori estimates (also for an upwind version). These upper and lower bounds measure the error in an energy (semi)norm between the exact solution and the numerical solution. The upper bound is robust in the sense that it does not depend on the variation of the model data. The lower bound, however, depends additionally on the local Péclet number. The local contributions of the a posteriori estimates are used to steer an adaptive mesh-refining algorithm. This strategy turns out to be very suitable for the numerical treatment of transmission problems, which have singularities or boundary/internal layers. Several numerical examples illustrate the effectiveness of the new conservative adaptive coupling method.