Algebraic multilevel preconditioner with projectors

Konstantin Lipnikov

T-7, MS-B284, Los Alamos, NM 87545

Yuri Kuznetsov

Department of Mathematics, University of Houston, TX 77204


The two-level algebraic preconditioner with projectors has been proposed in [1] for symmetric positive definite stiffness matrices. The underlying grid is partitioned into non-overlapping substructures of arbitrary shapes (grid subdomains, superelements or simply original grid cells). A projector to the kernel of a subdomain stiffness matrix is build for each subdomain. A coarse grid matrix is an approximation to the original stiffness matrix in the space of images of the projectors. If the original stiffness matrix is a finite element approximation of an elliptic operator, the coarse grid matrix describes connections between integral averages a discrete function over subdomains.

In the talk we shall prove the convergence of the two-level scheme for the case of elliptic operators. We shall show that the convergence rate is independent of jumps in coefficients between subdomains. A multilevel preconditioner based on the two-level scheme will be studied numerically for the case of highly distorted meshes and strongly varying coefficients. In comparison with a few other multilevel preconditioners, it demostrates robust behavior over a much bigger class of problems [2].

[1] Yu.Kuznetsov, Two-level preconditioners with projectors for unstructured grids. Russian J. Numer. Anal. Math. Modelling, (2000), v.15, pp.247-256.

[2] Yu.Kuznetsov, K.Lipnikov, Parallel numerical methods for the diffusion and Maxwell equations in heterogeneous media on strongly distorted meshes, Technical Report, University of Houston, May, 2002.