A Two-level Preconditioner for Anisotropic Diffusion Problems

O. Boyarkin, I. Kapyrin, Yu. Kuznetsov, and N. Yavich

Email: yavich@math.uh.edu
Postal address: Department of Mathematics, University of Houston, 651 Philip G. Hoffman Hall, Houston TX 77204-3008

In this talk, we present a new two-level preconditioner for algebraic systems arising from finite volume discretization of 3D anisotropic diffusion problems on prismatic meshes. With an appropriate choice of the coarse grid cell size, the condition  number of the preconditioned system is O(pnxy) where nxy is the number of cells in the xy-projection of the computational grid, i.e. the condition number does not depend on the mesh step size in the z-direction and jumps in the diffusion tensor and reaction term. This makes the new preconditioner very robust for problems with strong anisotropy. Theoretical investigation is illustrated with numerical experiments: the new preconditioner is compared with the z-line block Jacobi (BJ) preconditioner and the St¨uben’s algebraic multigrid (SAMG) preconditioner. Numerical experiments show that the preconditioned conjugate gradient method with the proposed preconditioner converges 5-20 times faster than with the BJ preconditioner and 2-6 times faster than with the SAMG preconditioner.