Block and asynchronous two-stage methods for mildly nonlinear systems

Abstract.

Block parallel iterative methods for the solution of mildly nonlinear systems of equations of the form $Ax=\Phi(x)$ are studied. Two-stage methods, where the solution of each block is approximated by an inner iteration, are treated. Both synchronous and asynchronous versions are analyzed, and both pointwise and blockwise convergence theorems provided. The case where there are overlapping blocks is also considered. The analysis of the asynchronous method when applied to linear systems includes cases not treated before in the literature.