Some Recent Advances in Multigrid Methods

Abstract

Publisher Summary Multigrid methods are fast iterative methods for the solution of large systems of algebraic equations. Most applications of multigrid and related methods involve the solving of sparse systems arising by the discretizations of partial differential equations. This chapter presents the basic form of the multigrid algorithm. It also presents methods based on the concept of decomposition of the space into a number of subspaces, where each subspace is the range of a prolongation mapping of a full rank from a lower dimensional space. The chapter also discusses multigrid in elasticity, multigrid for stokes equations, and eigenvalue problems. The application of multigrid principles to fluids as well as other multigrid developments is also represented in the chapter. There are several approaches to multigrid for nonlinear problems: multigrid iteration can be used as a linear solver for the linearized problem so that few multigrid iterations are made in each step of the multigrid method, or multigrid can be applied directly to the nonlinear problem so that the problems on all levels are nonlinear