Lecture 07: Nonlinear Preconditioning Methods and Applications

We consider solving system of nonlinear algebraic equations arising from the discretization of partial differential equations. Inexact Newton is a popular technique for such problems. When the nonlinearities in the system are well-balanced, Newton\u27s method works well, but when a small number of nonlinear functions in the system are much more nonlinear than the others, Newton may converge slowly or even stagnate. In such a situation, we introduce some nonlinear preconditioners to balance the nonlinearities in the system. The preconditioners are often constructed using a combination of some domain decomposition methods and nonlinear elimination methods. For the nonlinearly preconditioned problem, we show that fast convergence can be restored. In this talk we first review the basic algorithms, and then discuss some recent progress in the applications of nonlinear preconditioners for some difficult problems arising in computational mechanics including both fluid dynamics and solid mechanics

Restricted Additive Schwarz Preconditioners with Harmonic Overlap for Symmetric Positive Definite Linear Systems

A restricted additive Schwarz (RAS) preconditioning technique was introduced recently for solving general nonsymmetric sparse linear systems. In this paper, we provide one-level and two-level extensions of RAS for symmetric positive definite problems using the so-called harmonic overlaps (RASHO). Both RAS and RASHO outperform their counterparts of the classical additive Schwarz variants (AS). The design of RASHO is based on a much deeper understanding of the behavior of Schwarz-type methods in overlapping subregions and in the construction of the overlap. In RASHO, the overlap is obtained by extending the nonoverlapping subdomains only in the directions that do not cut the boundaries of other subdomains, and all functions are made harmonic in the overlapping regions. As a result, the subdomain problems in RASHO are smaller than those of AS, and the communication cost is also smaller when implemented on distributed memory computers, since the right-hand sides of discrete harmonic systems are always zero and therefore do not need to be communicated. We also show numerically that RASHO-preconditioned CG takes fewer iterations than the corresponding AS-preconditioned CG. A nearly optimal theory is included for the convergence of RASHO-preconditioned CG for solving elliptic problems discretized with a finite element method