The nested recursive two-level factorization method for nine-point difference matrices

• Published in: SIAM journal on scientific and statistical computing Vol. 12; no. 6; pp. 1373 - 1400
• Main Authors: AXELSSON, O, EIJKHOUT, V
• Format: Journal Article
• Language: English
• Published: Philadelphia, PA Society for Industrial and Applied Mathematics 01.11.1991
• Subjects: Approximation; Exact sciences and technology; Mathematics; Methods; Numerical analysis; Numerical analysis. Scientific computation; Numerical linear algebra; Polynomials; Sciences and techniques of general use; Stencils; Matrix element; Finite element method; Ordering; Iterative method; Eigenvalue problem; Factorization; Multilevel system
• ISSN: 0196-5204; 1064-8275; 2168-3417; 1095-7197
• DOI: 10.1137/0912075

Nested recursive two-level factorization methods for nine-point difference matrices are analyzed. Somewhat similar in construction to multilevel methods for finite element matrices, these methods use recursive red-black orderings of the meshes, approximating the nine-point stencils by five-point ones in the red points and then forming the reduced system explicitly. Because this Schur complement is again a nine-point matrix (on a skew grid this time), the process of approximating and factorizing can be applied anew. Progressing until a sufficiently coarse grid has been reached, this procedure gives a multilevel preconditioner for the original matrix. Solving the levels in $V$-cycle order will not give an optimal order method (that is, with a total work proportional to the number of unknowns), but we show that using certain combinations of $V$-cycles and $W$-cycles will give methods of both optimal order of numbers of iterations and computational complexity. Since all systems to be solved during a preconditioner solve are of diagonal form, the method is suitable for execution on massively parallel architectures.