The method of subspace corrections

This paper gives an overview for the method of subspace corrections. The method is first motivated by a discussion on the local behavior of high-frequency components in a solution to an elliptic problem. A simple domain decomposition method is discussed as an illustrative example and multigrid methods are discussed in more detail. Brief discussion are also given to some non-linear examples including eigenvalue problems, obstacle problems and liquid crystal modelings. The relationship between the method of subspace correction and the method of alternating projects is observed and discussed.