The transport of substances back and forth between surface water and groundwater is a very serious problem. We study herein the mathematical model of this setting consisting of the Stokes equations in the fluid region coupled with the Darcy equations in the porous medium, coupled across the interface by the Beavers-Joseph-Saffman conditions. We prove existence of weak solutions and give a complete analysis of a finite element scheme which allows a simulation of the coupled problem to be uncoupled into steps involving porous media and fluid flow subproblems. This is important because there are many "legacy" codes available which have been optimized for uncoupled porous media and fluid flow. Key words. coupled porous media and fluid flow, Stokes and Darcy equations, Beavers-JosephSaffman condition, weak solutions, finite element scheme, error estimates AMS subject classifications. 35Q35, 65N30, 65N15, 76D07, 76S05 PII. S00361429013927661. Introduction and the model. There are many serious problems currently facing the world in which the coupling between groundwater and surface water is important. These include questions such as predicting how pollution discharged into streams, lakes, and rivers makes its way into the water supply. This coupling is also important in technological applications involving filtration.The aim of our research is to begin the study of the following problem: an incompressible fluid in a region Ω 1 can flow both ways across an interface Γ I into a domain Ω 2 which is a porous medium saturated with the same fluid. The mathematical theory and numerical analysis of each subproblem is well developed, and reliable codes are available. Nevertheless, the mathematical theory of the coupled problem seems to be not completely understood. The model of this situation which is most accessible to large scale computations consists of the Navier-Stokes equations (or Stokes equations) in the fluid region coupled across an interface with the Darcy equations for the filtration velocity in the porous medium. This leads to mathematical difficulties arising from the coupled system of equations of different orders in different regions. See Jäger and Mikelić [16], Payne and Straughan [22] for the beginning of analytical studies of this problem. (For the Brinkman model of porous media flow this difficulty does not occur; see Jäger and Mikelić [17], Angot [1].) The second issue concerns the correct transmission conditions on the interface. The Beavers-Joseph-Saffman interface conditions [3, 25] are now well established. The third difficulty is technical: where the interface meets the other boundaries, there are incompatibilities between the imposed boundary conditions.

Abstract. We develop multiscale mortar mixed finite element discretizations for second order elliptic equations. The continuity of flux is imposed via a mortar finite element space on a course grid scale, while the equations in the coarse elements (or subdomains) are discretized on a fine grid scale. The polynomial degree of the mortar and subdomain approximation spaces may differ; in fact, the mortar space achieves approximation comparable to the fine scale on its coarse grid by using higher order polynomials. Our formulation is related to, but more flexible than, existing multiscale finite element and variational multiscale methods. We derive a priori error estimates and show, with appropriate choice of the mortar space, optimal order convergence and some superconvergence on the fine scale for both the solution and its flux. We also derive efficient and reliable a posteriori error estimators, which are used in an adaptive mesh refinement algorithm to obtain appropriate subdomain and mortar grids. Numerical experiments are presented in confirmation of the theory.

We present an expanded mixed finite element approximation of second-order elliptic problems containing a tensor coefficient. The mixed method is expanded in the sense that three variables are explicitly approximated, namely, the scalar unknown, the negative of its gradient, and its flux (the tensor coefficient times the negative gradient). The resulting linear system is a saddle point problem. In the case of the lowest order Raviart-Thomas elements on rectangular parallelepipeds, we approximate this expanded mixed method by incorporating certain quadrature rules. This enables us to write the system as a simple, cell-centered finite difference method requiring the solution of a sparse, positive semidefinite linear system for the scalar unknown. For a general tensor coefficient, the sparsity pattern for the scalar unknown is a 9-point stencil in two dimensions and 19 points in three dimensions. Existing theory shows that the expanded mixed method gives optimal order approximations in the L 2-and H −s-norms (and superconvergence is obtained between the L 2-projection of the scalar variable and its approximation). We show that these rates of convergence are retained for the finite difference method. If h denotes the maximal mesh spacing, then the optimal rate is O(h). The superconvergence rate O(h 2) is obtained for the scalar unknown and rate O(h 3/2) for its gradient and flux in certain discrete norms; moreover, the full O(h 2) is obtained in the strict interior of the domain. Computational results illustrate these theoretical results.

We develop a mixed finite element method for single phase flow in porous media that reduces to cell-centered finite differences on quadrilateral and simplicial grids and performs well for discontinuous full tensor coefficients. Motivated by the multipoint flux approximation method where sub-edge fluxes are introduced, we consider the lowest order Brezzi-Douglas-Marini (BDM) mixed finite element method. A special quadrature rule is employed that allows for local velocity elimination and leads to a symmetric and positive definite cell-centered system for the pressures. Theoretical and numerical results indicate second-order convergence for pressures at the cell centers and first-order convergence for sub-edge fluxes. Second-order convergence for edge fluxes is also observed computationally if the grids are sufficiently regular.

Abstract. We consider mixed finite element methods for second order elliptic equations on nonmatching multiblock grids. A mortar finite element space is introduced on the nonmatching interfaces. We approximate in this mortar space the trace of the solution, and we impose weakly a continuity of flux condition. A standard mixed finite element method is used within the blocks. Optimal order convergence is shown for both the solution and its flux. Moreover, at certain discrete points, superconvergence is obtained for the solution and also for the flux in special cases. Computational results using an efficient parallel domain decomposition algorithm are presented in confirmation of the theory.