Implicit Alternating Direction Methods

in general plane regions and with respect to linear boundary conditions, is a classical problem of numerical analysis. Many such boundary value problems have been solved successfully on high-speed computing machines, using the (iterative) Young-Frankel "successive overrelaxation" (SOR) method as defined in [l] and [2], and variants thereof ("line" and "block" overrelaxation). For this method, estimates of the rate of convergence and the "optimum relaxation factor" can both be rigorously extended from the special case of — \72u = S in a rectangle, and Dirichlet-type boundary conditions, to the general case. Recently, two variants [3; 4] of a new "implicit alternating direction" (IAD) method have been proved to converge much more rapidly, in the special case just mentioned, than the successive overrelaxation method and its variants. This fact has led to much speculation regarding the relative rates of convergence of SOR and IAD methods for more general elliptic boundary value problems. In [3, p. 41 ], success was reported in solving — S/2u = S for "several examples involving more complex regions and less simple boundary conditions," but no theoretical analysis was given of the convergence rate of the "Peaceman-Rachford" process used in these examples. In [4, p. 421 ], the "Douglas-Rachford" process was asserted to be stable (i.e., convergent) for — V2w = S and Dirichlet-type boundary conditions in general plane regions, but again the cases of variable D, S?^0, and mixed boundary conditions were not covered. Our main result below (Theorem 3) is that the convergence estimates given in [3] and [4] are applicable to the modified Helmholtz equation c2u — V2m = ,S in a rectangle with sides parallel to the axes, with the boundary condition du/dn+ku = 0, £2:0, and essentially to no other case. In dramatic contrast to the situation as regards SOR methods, the analysis of the rectangular case, via discrete Fourier analysis of eigenfunctions, gives no clue as to the general self-adjoint case. Nevertheless, IAD methods work well for many other cases of great interest. We present in §§8-9 also a few positive results, giving partial theoretical justification for this success.