Computational Methods in Applied Mathematics Vol. 12 (2012), No. 4, pp. 415–436 c ⃝ 2012 Institute of Mathematics, NAS of Belarus Doi: 10.2478/cmam-2012-0031 Robust Two-level Domain Decomposition Preconditioners for High-contrast Anisotropic Flows in Multiscale Media Yalchin Efendiev · Juan Galvis · Raytcho Lazarov · Svetozar Margenov · Jun Ren In memory of Sergei Nepomnyaschikh - a pioneer in domain decomposition methods Abstract — In this paper we discuss robust two-level domain decomposition precon- ditioners for highly anisotropic heterogeneous multiscale problems. We present a con- struction of several coarse spaces that employ standard finite element and multiscale basis functions and discuss techniques to reduce the dimensions of coarse spaces without sacrificing the robustness. We experimentally study the performance of the precondi- tioner on a variety two-dimensional test problems with channels of high anisotropy. The numerical tests confirm the robustness of the perconditioner with respect to the underlying physical parameters. 2010 Mathematical subject classification: 65N20; 65M25. Keywords: heterogeneous anisotropic media; domian decomposition; robust precondi- tioners; Schwarz method. 1. Introduction Anisotropy in the diffusion arises in many applications such as flow in porous media, heat transfer, and other areas in science and engineering. In porous media, high anisotropy can be due to the presence of fractures that can have preferred high-conductivity directions. Because of high variations among the matrix and fracture conductivities, the permeability can have Yalchin Efendiev Department of Mathematics, TAMU, College Station, TX 77843-3368, USA. E-mail: efendiev@math.tamu.edu Juan Galvis Department of Mathematics, TAMU, College Station, TX 77843, USA. and Departamento de Matem´aticas, Universidad Nacional de Colombia, Carrera 30, calle 45, Ciudad Universitaria Bogot´ a, Colombia E-mail: jcgalvisa@unal.edu.co Raytcho Lazarov Department of Mathematics, TAMU, College Station, TX 77843-3368, USA. E-mail: lazarov@math.tamu.edu Svetozar Margenov Institute of Information and Communication Technologies, Bulgarian Academy of Sciences, Acad. G. Bonchev St., Block 2, 1113 - Sofia, BULGARIA E-mail: margenov@parallel.bas.bg Jun Ren Department of Mathematics, TAMU, College Station, TX 77843-3368, USA. E-mail: junren@math.tamu.edu Brought to you by | University of Connecticut Authenticated Download Date | 6/2/15 5:45 PM