Abstract
This chapter describes the block-relaxation method for solution of the Dirichlet problem. It presents a new approach to the algorithmic implementation and optimization for one class of block-relaxation methods for the solution of the Dirichlet problem for elliptic equations. It does not obtain results in the most general form; possible generalizations are sufficiently evident, and the proofs require only some more complicated mathematical calculations. The chapter considers the problem Lu = f in Ω, u = 0 in ∂Ω, with a strongly elliptic operator L = L1 + L2 where L1 = −(∂P/∂x) (∂/∂x) and L2 = −(∂Q/ ∂y) (∂/∂y) in some two dimensional connected domain Ω with boundary ∂Ω such that Ω¯ is the union of a finite number of rectangles Ωii = 1, p¯.
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