Russ. J. Numer. Anal. Math. Modelling, Vol. 22, No. 4, pp. 377–391 (2007) DOI 10.1515/ RJNAMM.2007.018 c  de Gruyter 2007 Numerical analysis of a two-level preconditioner for the diffusion equation with an anisotropic diffusion tensor Yu. A. KUZNETSOV ∗ , O.V. BOIARKINE ∗ , I. V. KAPYRIN † , and N. B. YAVICH ∗ Abstract — We develop and analyze a new two-level preconditioner for algebraic systems arising from finite volume discretization of 3D anisotropic diffusion problems on prismatic meshes. The pre- conditioner is based on partitioning of the mesh in the (x, y)-plane into non-overlapping subdomains and on a special coarsening algorithm. The condition number of the preconditioned matrix does not depend on the coefficients in the diffusion operator. Numerical experiments confirm the theoretical re- sults and reveal the competitiveness of the new preconditioner with respect to the algebraic multigrid preconditioner. 1. Introduction Many engineering applications involve approximations of the diffusion equation with anisotropic coefficients on polyhedral meshes. The underlying algebraic sys- tems are known to be very ill-conditioned. In this paper we develop and analyze an essentially new approach originally proposed in [3] to construct two-level pre- conditioners for the diffusion equation. We consider the case of special polyhedral domains and special prismatic meshes. The diffusion tensor is assumed to be a diag- onal matrix, and the simplest version of the finite volume method is utilized for the discretization of the diffusion equation. The choice of the geometry of the domain and the meshes is motivated by applications in reservoir simulation. The paper is organized as follows. In Section 2 we describe the model problem and the matrices which arise in the simplest version of the finite volume method. In Section 3 we propose a new coarsening procedure based on partitioning of the mesh domain in the (x, y)-plane into non-overlapping subdomains. This procedure is a special modification of the algorithm earlier proposed in [2] and utilized in [4]. It can be proved [3] that the condition number of the preconditioned matrix does not depend on anisotropy in the diffusion tensor. ∗ University of Houston, Department of Mathematics, 4800 Calhoun Rd, Houston, Texas 77204, USA † Institute of Numerical Mathematics, Russian Academy of Sciences, Moscow 119333, Russia This work was supported by ExxonMobil Upstream Research Company. Brought to you by | New York University Bobst Library Technical Services Authenticated Download Date | 6/5/15 10:30 AM