Multiblock modeling of flow in porous media and applications

We investigate modeling flow in porous media in multiblock domain. Mixed finite element methods are used for subdomain discretizations. Physically meaningful boundary conditions are imposed on the non-matching interfaces via mortar finite element spaces. We investigate the pollution effect of nonmatching grids error on the numerical solution away from interfaces. We prove that most of the error in the velocity occurs along the interfaces, and that high accuracy is preserved in the interior of the subdomains. In case of discontinuous coefficients, the pollution from the singularity affects the accuracy in the whole domain. We investigate the upscaling error resulting when fine resolution data is approximated on a very coarse scale. Extending work of Wheeler and Yotov, we incorporate this upscaling error in an a posteriori error estimator for the pressure, velocity and mortar pressure. We employ a non-overlapping domain decomposition method reducing the global system to one that is solved iteratively via a preconditioned conjugate gradient method. This approach is suitable for parallel implementation. The balancing domain decomposition method for mixed finite elements following Cowsar, Mandel, and Wheeler is extended to the case of mortar mixed finite elements on non-matching multiblock grids. The algorithm involves solution of a mortar interface problem with one local Dirichlet solve and one local Neumann solve on each iteration. A coarse solve is used to guarantee consistency and to provide global exchange of information. Quasi-optimal condition number bounds independent of the jump in coefficients are derived. We finally consider multiscale mortar mixed finite element discretizations for single and two phase flows. We show optimal convergence and some superconvergence in the fine scale for the solution and its flux. We also derive efficient and reliable a posteriori error estimators suitable for adaptive mesh refinement. We have incorporated the above methods into a parallel multiblock simulator on unstructured prismatic meshes employing a non-overlapping domain decomposition algorithm and mortar spaces. Numerical experiments are presented confirming all theoretical results

Robust a posteriori error control and adaptivity for multiscale, multinumerics, and mortar coupling

International audienceWe consider discretizations of a model elliptic problem by means of different numerical methods applied separately in different subdomains, termed multinumerics, coupled using the mortar technique. The grids need not match along the interfaces. We are also interested in the multiscale setting, where the subdomains are partitioned by a mesh of size h , whereas the interfaces are partitioned by a mesh of much coarser size H , and where lower-order polynomials are used in the subdomains and higher-order polynomials are used on the mortar interface mesh. We derive several fully computable a posteriori error estimates which deliver a guaranteed upper bound on the error measured in the energy norm. Our estimates are also locally efficient and one of them is robust with respect to the ratio H/h under an assumption of sufficient regularity of the weak solution. The present approach allows bounding separately and comparing mutually the subdomain and interface errors. A subdomain/interface adaptive refinement strategy is proposed and numerically tested

Balancing domain decomposition for mortar mixed finite element methods

This paper deals with the problem of solving e#ciently the algebraic system arising in mortar mixed #nite element discretizations of elliptic equations. A non-overlapping domain decomposition algorithm developed for matching grids by Glowinski and Wheeler [12, 13] and later extended to non-matching grids [6, 14] is employed as a solver. The method reduces the global system to an interface problem which is symmetric and positive de#nite in the case of elliptic equations and can be solved iteratively via a preconditioned conjugate gradient method. This approach is very suitable for parallel implementation since the dominant cost is solving subdomain problems. The feasibility of the domain decomposition solver depends critically on the rate of convergence of the interface iteration and ultimately on the conditioning of the interface operator. The goal of this paper is to extend to the case of non-matching multiblock grids the balancing preconditioner for mixed #nite elements developed by Cowsar et al. [15]. Other substructuring preconditioners for mortar #nite elements can be found in References [16--18]. The balancing domain decomposition method was introduced by Mandel [19] for Galerkin #nite elements and later analysed by Mandel and Brezina [20]. The algorithm is based on the Neumann--Neumann preconditioner [21--23] and involves an iterative solution of the interface problem with one local Dirichlet solve (action of the operator) and one local Neumann solve (action of the preconditioner) per subdomain on each iteration. A coarse problem is added to guarantee that the Neumann problems are consistent which also provides global exchange of information across subdomains. The condition number analysis in Reference [15] pivots around a characterization of the interface biline..

A multiscale mortar mixed finite element method

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. Key words. Multiscale, mixed finite element, mortar finite element, error estimates, a posteriori, superconvergence, multiblock, non-matching grids AMS subject classifications. 65N06, 65N12, 65N15, 65N22, 65N3

A global Jacobian method for mortar discretizations of nonlinear porous media flows

We describe a non-overlapping domain decomposition algorithm for nonlinear porous media flows discretized with the multiscale mortar mixed finite element method. There are two main ideas: (1) linearize the global system in both subdomain and interface variables simultaneously to yield a single Newton iteration; and (2) algebraically eliminate subdomain velocities (and optionally, subdomain pressures) to solve linear systems for the 1st (or the 2nd) Schur complements. Solving the 1st Schur complement system gives the multiscale solution without the need to solve an interface iteration. Solving the 2nd Schur complement system gives a linear interface problem for a nonlinear model. The methods are less complex than a previously developed nonlinear mortar algorithm, which requires two nested Newton iterations and a forward difference approximation. Furthermore, efficient linear preconditioners can be applied to speed up the iteration. The methods are implemented in parallel, and a numerical study is performed to demonstrate convergence behavior and parallel efficiency