2006 MCS Divisional Seminars & Colloquia

Multiblock modeling of flow in porous media and applications

   Gergina Pencheva

 University of Pittsburgh

  Hosted by  Mihai Anitescu

10:30 AM, February 21, 2006
Building 221,  Room A216

We study second order elliptic equations which, in porous medium applications, model single phase flow. We solve the problem using multiblock domain decomposition methodology. 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. Several aspects of the resulting system are considered.

We develop and implement an efficient preconditioner that speeds up the domain decomposition solver in the case of mortar mixed finite elements on non-matching multiblock grids. The algorithm involves an iterative solution of a mortar interface problem with one local Dirichlet solve and one local Neumann solve per subdomain on each iteration. A coarse solve is used to guarantee that the Neumann problems are consistent and to provide global exchange of information across subdomains. Quasi-optimal condition number bounds are derived, which are independent of the jump in coefficients between subdomains. Numerical experiments confirming the theoretical results are presented.

We next investigate the pollution effect of nonmatching grids error on the numerical solution away from interfaces. We prove that in the case of nonmatching grids and smooth solutions most of the error occurs along the interfaces, and that high accuracy is preserved in the interior of the subdomains.

We also investigate the upscaling error resulting when fine resolution data is approximated at a very coarse scale. We prove explicit a posteriori error estimators for the pressure, velocity and mortar pressure errors that incorporate this upscaling error.

We finally consider 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 coarse scale, while the equations in the coarse elements (subdomains) are discretized on a fine scale. The polynomial degrees of approximations in the mortar space and in the subdomain spaces may be different. We derive error estimates and show, with appropriate choice of the mortar space, optimal convergence and some superconvergence in the fine scale for both the solution and its flux. We also derive several 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.