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In this paper, we consider the Neumann boundary value problem for the 3D diffusion equation with a diagonal diffusion tensor in a rectangular domain. The problem is discretized on rectangular meshes by the simplest version of the finite volume method. We assume that the coefficients of the discrete problem are dominant in the ‘horizontal’  xy -direction. A new approach based on earlier publications [5-7] for the construction of a multilevel preconditioner is proposed and analyzed. The main idea of the approach is to coarsen the discrete problem only in  xy -direction, keeping the mesh in  z -direction unchanged. We prove that the proposed preconditioner is spectrally equivalent to the system matrix and that the arithmetical complexity of the PCG method with this preconditioner is almost optimal. Numerical results are given.

Multilevel substructuring preconditioners for anisotropic diffusion problems on rectangular meshes