In this dissertation, we develop new multilevel approaches to precondition algebraic problems stemming from the finite volume discretization of the diffusion equation with anisotropic, discontinuous coefficients. Two approaches are discussed.;In the first approach, preconditioners are based on a partitioning of the mesh in the (x, y)-plane into non-overlapping subdomains and on a special coarsening algorithm within each of the mesh layers. We show that the technique can be directly applied to the prismatic meshes based on arbitrary 2D Voronoi mesh cells. Obtained numerical results comply with the theoretical statement that the condition number of the preconditioned system receives neither an impact from diffusion tensor anisotropy nor from the thickness of thin mesh layers.;The second approach is of the algebraic multigrid type. The approach is based on the representation of the graph of the system matrix as a union of clusters. The clusters are used to design and analyze coarsening algorithms. The multilevel framework involves inner Chebyshev iterations. We construct preconditioners for two model problems (domain with a thin layer and domain with a canal) and compare performance of our preconditioners with that of another algebraic multigrid preconditioner.
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