Overview

Multilevel Preconditioners: This new class of preconditioners can be viewed as one cycle of a standard multigrid method without the smoothing operations. They use the multigrid principle to capture the different length scales of the solution but rely on the conjugate gradient method to deal with other convergence difficulties. They offer the efficiency of multigrid methods and the robustness of the conjugate gradient method.

Tony F. Chan . Hierarchical Algorithms and Architectures for Parallel Scientific Computing . CAM report 90-10 (1990)

In many applications, there is a natural partitioning of the given matrix it is of importance to examine if the corresponding system can be solved more efficiently. For large-scale problems, iterative solution methods are usually more efficient than direct solution methods. The most important aspect of the iterative solution method is the choice of the preconditioning matrix.

Owe Axelsson. Iterative Solution Methods . Cambridge University Press (1994)

As shown in Section 2.1.2, the Schur complement matrix

(7.1)

obtained from eliminating the bottom left block of a 2 2 block matrix

(7.2)

naturally occurs during the LU decomposition (2.6). When A is SPD, the condition number of S is less than A (Section 2.1.2) and also that of a block diagonal preconditioned matrix [345]. In the general case, computing an exact Schur complement amounts to implementing the LU decomposition while approximating a Schur complement is equivalent to obtaining a special ILU decomposition (preconditioner). However, powerful preconditioners can be obtained by...