Elizabeth H. Cuthill and Richard S. Varga, A method of normalized block iteration, J. Assoc. Comput. Math. 6 (1959), 236-244. MR 22, 8651. Zbl. 88, p. 94.
1. Introduction
The Young-Frankel successive point overrelaxation scheme [14,4] has been shown [14] to be applicable in solving partial difference equations of elliptic type arising from discrete approximations to general partial differential equations of elliptic type. More recently, Arms, Gates, and Zondek [1] generalized the successive point overrelaxation scheme of Young-Frankel to what is called the successive block overrelaxation scheme, and they stated that, with certain additional assumptions, a theoretical advantage in the rates of convergence is always obtained in using successive block overrelaxation rather than successive point overrelaxation. In particular, for the numerical solution of the Dirichlet problem of uniform mesh size h in a rectangle, the successive block overrelaxation scheme [1] is asymptotically faster by a factor of 21/2 than the successive point overrelaxation shceme, as h --> 0. Despite that advantage, the successive block overrelaxation scheme has not been widely used, mainly because the usual computing machine application of block overrelaxation requires more arithmetic operations than point overrelaxation, and this increase in the number of arithmetic operations would appear to cancel any gains in the rates of convergence.