Pressure jump interface law for the Stokes-Darcy coupling: Confirmation by direct numerical simulations

It is generally accepted that the effective velocity of a viscous flow over a porous bed satisfies the Beavers-Joseph slip law. To the contrary, interface law for the effective stress has been a subject of controversy. Recently, a pressure jump interface law has been rigorously derived by Marciniak-Czochra and Mikeli\'c. In this paper, we provide a confirmation of the analytical result using direct numerical simulation of the flow at the microscopic level.Comment: 25 pages, preprin

Design of Numerical Methods for Simulating Models of a Solid Oxide Fuel Cell

The performance of fuel cells is significantly affected by “loss mechanisms”. This work is devoted to developing concepts for the efficient numerical computation of the diffusion polarization in the porous anode of a solid oxide fuel cell (SOFC). The following topics were covered: The first part of this work is focused on the numerical verification of coupling conditions for effective viscous flows over a porous medium. It is generally accepted that the “Beavers-Joseph-Saffman slip law” holds true for a main flow direction which is tangential to the interface. However, the interface law for the effective stress has been a subject of controversy. We provide a confirmation of the “pressure jump law”, which has been recently derived by Marciniak-Czochra and Mikelic, for a range of configurations using a direct numerical simulation of the flow at the microscopic level. The second part of this work is about the derivation of a goal-oriented, a posteriori error estimator for the finite element approximation of elliptic homogenization problems based on the “Dual Weighted Residual method” of Becker and Rannacher. In general, the solution of the macroscopic equation in the homogenized model depends on effective coefficients which in turn depend on the solutions of some additional auxiliary equations. Therefore, the accuracy of the physical goal functional is influenced by the discretization error of the macroscopic and the auxiliary solutions. By employing the error estimator developed in this work we can estimate the contribution of the discretization of each sub-problem (effective model and auxiliary problems) onto the overall error. These contributions are then balanced within a successive refinement cycle to set up an efficient discretization. Local error indicators are used to steer an adaptive mesh refinement for the macroscopic problem as well as the auxiliary problems. We demonstrate the functionality of this algorithm on some prototypical homogenization problems and on an effective model developed in this work to simulate the gas transport in the anode of an SOFC. In the latter, the diffusion polarization is the quantity of interest. For a given accuracy, the application of the local mesh refinement based on the adaptive algorithm in this context decreases the number of degrees of freedom and computation time significantly compared to the global mesh refinement