A rapidly convergent method for the inversion of separable, positive, self-adjoint discrete elliptic operators in three or more dimensions
type: Item
instance of: scholarly article
title: A rapidly convergent method for the inversion of separable, positive, self-adjoint discrete elliptic operators in three or more dimensions
zbMATH Open document ID: 0646.65049
DOI: 10.1016/0021-9991(87)90096-9
author: Eric D. Siggia
author: A. Pumir
published in: Journal of Computational Physics
publication date: 1987-01-01T00:00:00Z
review text: The authors observe in this brief note that if the operators \(L_ x\), \(L_ y\), \(L_ z\) have positive eigenvalues and commute, then the problem \((L_ x+L_ y+L_ z)\phi =\rho\) can be solved by the iteration \((L_ x+L_ y)\phi^{k+1}=\rho -L_ z\phi \quad k,\) \((L_ y+L_ z)\phi^{k+2}=\rho -L_ x\phi^{k+1},\quad (L_ z+L_ x)\phi^{k+3}=\rho -L_ y\phi^{k+2}.\) Analysis of each component of the eigenfunction expansion shows \(| \phi^{k+3}-\phi \quad k| \leq | \phi \quad k-\phi^{k-3}| /8.\) The authors remark that the case of a zero eigenvalue can be handled, and also that the method can be used as an inner iteration in an obvious extension to four dimensions and higher.
Mathematics Subject Classification ID: 65J10
Mathematics Subject Classification ID: 65Z05
Mathematics Subject Classification ID: 47F05
Mathematics Subject Classification ID: 47A50
Mathematics Subject Classification ID: 47A70
Mathematics Subject Classification ID: 35J05
Mathematics Subject Classification ID: 35P10
zbMATH DE Number: 4054936
zbMATH Keywords: separable, positive, self-adjoint discrete elliptic operators
zbMATH Keywords: Poisson equation
zbMATH Keywords: positive eigenvalues
zbMATH Keywords: eigenfunction expansion
zbMATH Keywords: inner iteration
reviewed by: James R. Kuttler
MaRDI profile type: Publication
full work available at URL: 0021-9991(87)90096-9
OpenAlex ID: W2019651071
cites work: Multi-Level Adaptive Solutions to Boundary-Value Problems
cites work: A direct Method for the Discrete Solution of Separable Elliptic Equations
cites work: On Direct Methods for Solving Poisson’s Equations
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description: scientific article; zbMATH DE number 4054936