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A mathematical and numerical model describing chemical transport in a StokesDarcy flow system is discussed. The flow equations are solved through domain decomposition using classical finite element methods in the Stokes region and mixed finite element methods in the Darcy region. The local discontinuous Galerkin (LDG) method is used to solve the transport equation. Models dealing with coupling between Stokes and Darcy equations have been extensively discussed in the literature. This paper focuses on the approximation of the transport equation. Stability of the LDG scheme is analyzed, and an a priori error estimate is proved. Several numerical examples verifying the theory and illustrating the capabilities of the method are presented.

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A multiscale flux basis algorithm is developed for the Stokes-Darcy flow problem. The method is based on a non-overlapping domain decomposition algorithm, where the global problem is reduced to a coarse scale mortar interface problem that is solved by an iterative solver. Subdomain solves are required at each interface iteration, so the cost for the method without a multiscale basis can be high when the number of subdomains or the condition number of the interface problem is large. The proposed algorithm involves precomputing a multiscale flux basis, which consists of the flux (or velocity trace) response from each mortar degree of freedom. It is computed by each subdomain independently before the interface iteration begins. The subdomain solves required at each iteration are substituted by a linear combination of the multiscale basis. This may lead to a significant reduction in computational cost since the number of subdomain solves is fixed, depending only on the number of mortar degrees of freedom associated with a subdomain. Several numerical examples are carried out to demonstrate the efficiency of the multiscale flux basis implementation for large-scale Stokes-Darcy problems.