A Robin-Type Domain Decomposition Method for a Novel Mixed-Type DG Method for the Coupled Stokes--Darcy Problem

Abstract

In this paper, we first propose and analyze a novel mixed-type DG method for the coupled Stokes-Darcy problem on simplicial meshes. The proposed formulation is locally conservative. A mixed-type DG method in conjunction with the stress-velocity formulation is employed for the Stokes equations, where the symmetry of stress is strongly imposed. The staggered DG method is exploited to discretize the Darcy equations. As such, the discrete formulation can be easily adapted to account for the Beavers-Joseph-Saffman interface conditions without introducing additional variables. Importantly, the continuity of normal velocity is satisfied exactly at the discrete level. A rigorous convergence analysis is performed for all the variables. Then we devise and analyze a domain decomposition method via the use of Robin-type interface boundary conditions, which allows us to solve the Stokes subproblem and the Darcy subproblem sequentially with low computational costs. The convergence of the proposed iterative method is analyzed rigorously. In particular, the proposed iterative method also works for very small viscosity coefficient. Finally, several numerical experiments are carried out to demonstrate the capabilities and accuracy of the novel mixed-type scheme, and the convergence of the domain decomposition method.