Inconsistencies and S.O.R. Convergence for the Discrete Neumann Problem

The behaviour of S.O.R. iterations for linear equations AU =Bis described for the case when A is singular. Small inconsistenciesmay arise in practical application, for example in systems derivedfrom Neumann problems. In that case the S.O.R. iterations donot converge. A simple transformation is presented under whichthe iterations converge to an approximate solution of AU = B,provided that the singularity of A is of a simple type. A practicalway of measuring the appropriate convergence rate is also described.For problems with property (A) and consistent ordering the optimumacceleration parameter is unaffected by the simple singularityof A. The behaviour of the iterations when A has singularitiesof general type is also described.