On the convergence rate of a space decomposition method
An abstract minimization problem for a \(C^2\)-functional \(F:V\to{\mathbb R}\) whose derivative \(F'\) is Lipschitz continuous and uniformly monotone on each bounded set of a reflexive Banach space \(V\) is considered. A space-decomposition method, relying on a decomposition \(V=V_1+\cdots +V_m\), is scrutinized in connection with the iterative minimization algorithm making partial minimizations on each subspaces \(V_1\), \dots , \(V_m\) whose minimizers are further used to determine a direction for a line-search step to up-date the current iteration. The rate of convergence is proved. It is mentioned that it is a unified framework for both domain decomposition method and multigrid method used in numerical solutions of boundary-value problems for elliptic partial differential equations.
scientific article; zbMATH DE number 1885787