Iterative Methods by Space Decomposition and Subspace Correction

The main purpose of this paper is to give a systematic introduction to a number of iterative methods for symmetric positive definite problems. Based on results and ideas from various existing works on iterative methods, a unified theory for a diverse group of iterative algorithms, such as Jacobi and Gauss–Seidel iterations, diagonal preconditioning, domain decomposition methods, multigrid methods, multilevel nodal basis preconditioners and hierarchical basis methods, is presented. By using the notions of space decomposition and subspace correction, all these algorithms are classified into two groups, namely parallel subspace correction (PSC) and successive subspace correction (SSC) methods. These two types of algorithms are similar in nature to the familiar Jacobi and Gauss–Seidel methods, respectively. A feature of this framework is that a quite general abstract convergence theory can be established. In order to apply the abstract theory to a particular problem, it is only necessary to specify a decomposition of the underlying space and the corresponding subspace solvers. For example, subspaces arising from the domain decomposition method are associated with subdomains whereas with the multigrid method subspaces are provided by multiple “coarser” grids. By estimating only two parameters, optimal convergence estimations for a given algorithm can be obtained as a direct consequence of the abstract theory.