. The rate of convergence of the Balancing Domain Decomposition method applied to the mixed finite element discretization of second order elliptic equations is analyzed. The Balancing Domain Decomposition method, introduced by Mandel in [24], is a substructuring method that involves at each iteration the solution of a local problem with Dirichlet data, a local problem with Neumann data, and a "coarse grid" problem to propagate information globally and to insure the consistency of the Neumann problems. It is shown that the condition number grows at worst like the logarithm squared of the ratio of the subdomain size to the element size, in both two and three dimensions and for elements of arbitrary order. The bounds are uniform with respect to coefficient jumps of arbitrary size between subdomains. The key component of our analysis is the demonstration of an equivalence between the norm induced by the bilinear form on the interface and the H 1=2 -norm of an interpolant of the boundary ...

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 semideflnite linear system for the scalar unknown. For a general tensor coefficient, the sparsity pattern for the scalar unknown is a nine point stencil in two dimensions, and 19 points in three dimensions. Existing theory shows that the expanded mixed method gives optimal order ap- proximations in the L a and H-S-norms (and superconvergence is obtained between the La-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 ) is obtained for the scalar unknown and rate O(h 3/) for its gradient and flux in certain discrete norms; moreover, the full O(h ) is obtained in the strict interior of the domain. Computational results illustrate these theoretical results.

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.

Abstract. Two-level additive Schwarz preconditioners are developed for the nonconforming P1 finite element approximation of scalar second-order symmetric positive definite elliptic boundary value problems, the Morley finite element approximation of the biharmonic equation, and the divergence-free nonconforming P1 finite element approximation of the stationary Stokes equations. The condition numbers of the preconditioned systems are shown to be bounded independent of mesh sizes and the number of subdomains in the case of generous overlap. 1.

We present an expanded mixed finite element method for solving second-order elliptic partial di#erential equations on geometrically general domains. For the lowest-order Raviart-Thomas approximating spaces, we use quadrature rules to reduce the method to cell-centered finite di#erences, possibly enhanced with some face-centered pressures. This substantially reduces the computational complexity of the problem to a symmetric, positive definite system for essentially only as many unknowns as elements. Our new method handles general shape elements (triangles, quadrilaterals, and hexahedra) and full tensor coefficients, while the standard mixed formulation reduces to finite di#erences only in special cases with rectangular elements. As in other mixed methods, we maintain the local approximation of the divergence (i.e., local mass conservation). In contrast, Galerkin finite element methods facilitate general element shapes at the cost of achieving only global mass conservation. Our method i...

This paper reviews and extends the theory and application of mimetic finite difference methods for the solution of diffusion problems in strongly heterogeneous non-isotropic materials. These difference operators satisfy the fundamental identities, conservation laws and theorems of vector and tensor calculus on nonorthogonal, nonsmooth, structured and unstructured computational grids. We provide explicit approximations for equations in two dimensions with discontinuous nondiagonal diffusion tensors. We mention the similarities and differences between the new methods and mixed finite element or hybrid mixed finite element methods.

We consider mixed finite element discretizations of linear second order elliptic boundary value problems with respect to an adaptively generated hierarchy of possibly highly nonuniform simplicial triangulations. In particular, we present and analyze four different kinds of error estimators: a residual based estimator, a hierarchical one, error estimators relying on the solution of local subproblems and on a superconvergence result, respectively. Finally, we examine the relationship between the presented error estimators and compare their local components.

. The purpose of this paper is to establish an equivalence between mixed and nonconforming finite element methods for second order elliptic problems on both triangular and rectangular finite elements in IR 2 and IR 3 , and to provide an analysis of multigrid methods for both methods based on the equivalence. We first show that the linear system arising from the mixed method can be algebraically condensed to a symmetric and positive definite system for Lagrange multipliers using features of mixed finite element spaces and that the system for the Lagrange multipliers is identical to the system arising from the nonconforming method. Then we prove that optimal order multigrid algorithms can be developed for both methods. Two types of multigrid methods are considered in this paper. The first one makes use of the coarse-grid correction on nonconforming finite element spaces, while the second one has the coarse-grid correction step established on conforming finite element spaces. Finally,...

. In this paper domain decomposition algorithms for mixed finite element methods for linear and quasilinear second order elliptic problems in IR 2 and IR 3 are developed. A convergence theory for two-level and multilevel Schwarz methods applied to the algorithms under consideration is given, and its extension to other substructuring methods such as vertex space and balancing domain decomposition methods is considered. It is shown that the condition number of these iterative methods is bounded uniformly from above in the same manner as in the theory of domain decomposition methods for conforming and nonconforming finite element methods for the same differential problems. Numerical experiments are presented to illustrate the present techniques. 1. Introduction. This is the second paper of a sequence where we develop and analyze efficient iterative algorithms for solving the linear system arising from mixed finite element methods for linear and quasilinear second order elliptic proble...

. A new approach of constructing algebraic multilevel preconditioners for mixed finite element methods for second order elliptic problems with tensor coefficients on general geometry is proposed. The linear system arising from the mixed methods is first algebraically condensed to a symmetric, positive definite system for Lagrange multipliers, which corresponds to a linear system generated by standard nonconforming finite element methods. Algebraic multilevel preconditioners are then constructed for this system based on a triangulation of parallelepipeds into tetrahedral substructures. Explicit estimates of condition numbers and simple computational schemes are established for the constructed preconditioners. Finally, numerical results for the mixed finite element methods are presented to illustrate the present theory. Key words. mixed method, nonconforming method, multilevel preconditioner, condition number, second order elliptic problem AMS(MOS) subject classifications. 65N30, 65N22...