Non-iterative numerical solution of boundary-value problems

Why it is Difficult to Solve Helmholtz Problems with Classical Iterative Methods

“…The first such techniques [44,58,11,55], also known as domain imbedding or capacitance matrix methods, were developed to extend the efficiency of fast Poisson solvers based on the Fast Fourier Transform or cyclic reduction also to problems for which these methods are not directly applicable, as they require some form of separation of variables. In [21] (see also [22]) this idea was applied to exterior boundary value problems for the Helmholtz equation in two dimensions, and it was shown how the Sommerfeld radiation condition can be incorporated into a fast Poisson solver.…”

An Abridged Block Method for the Solution of the Dirichlet Problem for the Laplace Difference Equation

“…In the case where a rectangle has only one inner point, equation (3) reduces to the condition that the function be the mean of Its values at the adjacent points. The proof of the eXIstence and umqueness of the solution is an immediate consequence of the classical theorems for thIS problem [5] …”

“…(The corresponding formula for the three dimensional case was gIven by N. Wiener and H. B. Phillips [1]. A special case of the formula for two dimensions was developed by W. H. McCrea and F. J. W. Whipple [2] and the general formula for the two dimensional case was given by M. A. Hyman [3].) (See also [8] and [9])…”

Matrix decomposition algorithms for elliptic boundary value problems: a survey

“…In 1965, Hockney [112] introduced a method which combines cyclic reduction with an MDA. He claimed that the method of [150] was essentially that proposed by Hyman [116] in 1951. Subsequent developments of MDAs for finite difference methods are described in [61,79,121,162,188,189,197].…”