Über eine Methode zur numerischen Lösung der Poissonschen Differenzengleichung für beliebige Gebiete

Rank reduction and bordered inversion

“…We just obtained that one rank reduction step on matrix A −1 results in the inverse of the leading principal submatrix E. Relation (6.3) gives a direct connection between the rank reduction and the bordered inversion. It was first discovered by Egerváry [4] in the scalar case (H ∈ R), who developed it for solving difference equations with modified boundary conditions [5]. The same idea appears in Brezinski et al [2], who give credit to Duncan and call it reverse bordered inversion.…”

Direct solution of partial difference equations by tensor product methods

“…In the next section general second order separable partial difference equations are defined, analyzed, and an explicit expression for the solution of boundary value problems is obtained. Finally, it is shown how to efficiently evaluate the explicit form of the solution (one does not, as proposed in [12], compute the inverse of the matrix in (IA) even though this inverse is explicitly available). Several types of possible generalizations are investigated in Section 4.…”

“…Since the results reported in [5] were obtained, we have found that EGERVARY [12] applied tensor products to the analysis of the fivepoint approximation of Poisson's equation. His computational scheme is not feasible for large problems.…”

Matrix decomposition algorithms for elliptic boundary value problems: a survey

“…In these methods, the diagonalization is performed in both variables so that in Step 2 of the algorithm the systems are diagonal. According to [150], tensor product methods of this type were developed in 1960 in [82] for the five-point finite difference approximation of (1.5) but an inefficient solution procedure was formulated. In 1965, Hockney [112] introduced a method which combines cyclic reduction with an MDA.…”