Analysis and Application of Multigrid Preconditioners for Singularly Perturbed Boundary Value Problems
This paper analyzes multigrid preconditioners designed to accelerate the convergence of various singularly perturbed boundary value problems for which the coefficients of low-order derivative terms are multiplied by a large parameter K. The multigrid preconditioner is not applied to the given operat...
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| Veröffentlicht in: | SIAM journal on numerical analysis 1989-10, Vol.26 (5), p.1090-1123 |
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| container_title | SIAM journal on numerical analysis |
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| creator | Goldstein, C. I. |
| description | This paper analyzes multigrid preconditioners designed to accelerate the convergence of various singularly perturbed boundary value problems for which the coefficients of low-order derivative terms are multiplied by a large parameter K. The multigrid preconditioner is not applied to the given operator, but rather to a simpler operator (such as the Laplacian) for which its implementation can be carried out much more efficiently. The preconditioner is shown to be effective for all K ≥ 0 provided the coarsest grid size is suitably chosen in terms of K. The multigrid algorithm does not solve the coarse-grid equations but consists of, at most, a few relaxation sweeps on this grid level. The main results are proved for a W cycle and a "variable" V cycle using Fourier analysis. Numerical experiments indicate the validity of these results for a standard V cycle as well. |
| doi_str_mv | 10.1137/0726061 |
| format | Article |
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I.</creator><creatorcontrib>Goldstein, C. I.</creatorcontrib><description>This paper analyzes multigrid preconditioners designed to accelerate the convergence of various singularly perturbed boundary value problems for which the coefficients of low-order derivative terms are multiplied by a large parameter K. The multigrid preconditioner is not applied to the given operator, but rather to a simpler operator (such as the Laplacian) for which its implementation can be carried out much more efficiently. The preconditioner is shown to be effective for all K ≥ 0 provided the coarsest grid size is suitably chosen in terms of K. The multigrid algorithm does not solve the coarse-grid equations but consists of, at most, a few relaxation sweeps on this grid level. The main results are proved for a W cycle and a "variable" V cycle using Fourier analysis. Numerical experiments indicate the validity of these results for a standard V cycle as well.</description><identifier>ISSN: 0036-1429</identifier><identifier>EISSN: 1095-7170</identifier><identifier>DOI: 10.1137/0726061</identifier><identifier>CODEN: SJNAEQ</identifier><language>eng</language><publisher>Philadelphia, PA: Society for Industrial and Applied Mathematics</publisher><subject>Applied mathematics ; Boundary value problems ; Eigenvalues ; Eigenvectors ; Exact sciences and technology ; Fourier analysis ; Initial guess ; Integers ; Iterative methods ; Mathematics ; Methods ; Multigrid methods ; Numerical analysis ; Numerical analysis. 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I.</creatorcontrib><title>Analysis and Application of Multigrid Preconditioners for Singularly Perturbed Boundary Value Problems</title><title>SIAM journal on numerical analysis</title><description>This paper analyzes multigrid preconditioners designed to accelerate the convergence of various singularly perturbed boundary value problems for which the coefficients of low-order derivative terms are multiplied by a large parameter K. The multigrid preconditioner is not applied to the given operator, but rather to a simpler operator (such as the Laplacian) for which its implementation can be carried out much more efficiently. The preconditioner is shown to be effective for all K ≥ 0 provided the coarsest grid size is suitably chosen in terms of K. The multigrid algorithm does not solve the coarse-grid equations but consists of, at most, a few relaxation sweeps on this grid level. The main results are proved for a W cycle and a "variable" V cycle using Fourier analysis. Numerical experiments indicate the validity of these results for a standard V cycle as well.</description><subject>Applied mathematics</subject><subject>Boundary value problems</subject><subject>Eigenvalues</subject><subject>Eigenvectors</subject><subject>Exact sciences and technology</subject><subject>Fourier analysis</subject><subject>Initial guess</subject><subject>Integers</subject><subject>Iterative methods</subject><subject>Mathematics</subject><subject>Methods</subject><subject>Multigrid methods</subject><subject>Numerical analysis</subject><subject>Numerical analysis. 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I.</author></sort><facets><frbrtype>5</frbrtype><frbrgroupid>cdi_FETCH-LOGICAL-c233t-deedb4f68909023145f34062ce305d33d6a7c55c9b95efd32d138198f2c322643</frbrgroupid><rsrctype>articles</rsrctype><prefilter>articles</prefilter><language>eng</language><creationdate>1989</creationdate><topic>Applied mathematics</topic><topic>Boundary value problems</topic><topic>Eigenvalues</topic><topic>Eigenvectors</topic><topic>Exact sciences and technology</topic><topic>Fourier analysis</topic><topic>Initial guess</topic><topic>Integers</topic><topic>Iterative methods</topic><topic>Mathematics</topic><topic>Methods</topic><topic>Multigrid methods</topic><topic>Numerical analysis</topic><topic>Numerical analysis. Scientific computation</topic><topic>Partial differential equations, boundary value problems</topic><topic>Preconditioning</topic><topic>Sciences and techniques of general use</topic><toplevel>peer_reviewed</toplevel><toplevel>online_resources</toplevel><creatorcontrib>Goldstein, C. I.</creatorcontrib><collection>Pascal-Francis</collection><collection>CrossRef</collection><collection>ProQuest Central (Corporate)</collection><collection>Access via ABI/INFORM (ProQuest)</collection><collection>ABI/INFORM Global (PDF only)</collection><collection>Agricultural Science Collection</collection><collection>ProQuest Central (purchase pre-March 2016)</collection><collection>ABI/INFORM Global (Alumni Edition)</collection><collection>Biology Database (Alumni Edition)</collection><collection>Military Database (Alumni Edition)</collection><collection>Science Database (Alumni Edition)</collection><collection>Telecommunications (Alumni Edition)</collection><collection>Computing Database (Alumni Edition)</collection><collection>ProQuest SciTech Collection</collection><collection>ProQuest Technology Collection</collection><collection>ProQuest Natural Science Collection</collection><collection>ProQuest Central (Alumni) (purchase pre-March 2016)</collection><collection>ABI/INFORM Collection (Alumni Edition)</collection><collection>Research Library (Alumni Edition)</collection><collection>Materials Science & Engineering Collection</collection><collection>ProQuest Central (Alumni Edition)</collection><collection>ProQuest Central UK/Ireland</collection><collection>Advanced Technologies & Aerospace Collection</collection><collection>Agricultural & Environmental Science Collection</collection><collection>ProQuest Central Essentials</collection><collection>Biological Science Collection</collection><collection>ProQuest Central</collection><collection>Business Premium Collection</collection><collection>Technology Collection</collection><collection>Natural Science Collection</collection><collection>ProQuest One Community College</collection><collection>ProQuest Materials Science Collection</collection><collection>ProQuest Central Korea</collection><collection>Business Premium Collection (Alumni)</collection><collection>ABI/INFORM Global (Corporate)</collection><collection>ProQuest Central Student</collection><collection>Research Library Prep</collection><collection>SciTech Premium Collection</collection><collection>ProQuest Computer Science Collection</collection><collection>ProQuest Business Collection (Alumni Edition)</collection><collection>ProQuest Business Collection</collection><collection>Computer Science Database</collection><collection>Materials Science Database</collection><collection>ABI/INFORM Professional Advanced</collection><collection>ProQuest Engineering Collection</collection><collection>ProQuest Biological Science Collection</collection><collection>ABI/INFORM Global</collection><collection>Agricultural Science Database</collection><collection>Computing Database</collection><collection>Military Database</collection><collection>Research Library</collection><collection>Science Database</collection><collection>Telecommunications Database</collection><collection>Biological Science Database</collection><collection>Engineering Database</collection><collection>Research Library (Corporate)</collection><collection>Advanced Technologies & Aerospace Database</collection><collection>ProQuest Advanced Technologies & Aerospace Collection</collection><collection>Environmental Science Database</collection><collection>Materials Science Collection</collection><collection>One Business (ProQuest)</collection><collection>ProQuest One Business (Alumni)</collection><collection>ProQuest One Academic Eastern Edition (DO NOT USE)</collection><collection>ProQuest One Academic</collection><collection>ProQuest One Academic UKI Edition</collection><collection>Engineering Collection</collection><collection>Environmental Science Collection</collection><collection>ProQuest Central Basic</collection><jtitle>SIAM journal on numerical analysis</jtitle></facets><delivery><delcategory>Remote Search Resource</delcategory><fulltext>fulltext</fulltext></delivery><addata><au>Goldstein, C. I.</au><format>journal</format><genre>article</genre><ristype>JOUR</ristype><atitle>Analysis and Application of Multigrid Preconditioners for Singularly Perturbed Boundary Value Problems</atitle><jtitle>SIAM journal on numerical analysis</jtitle><date>1989-10-01</date><risdate>1989</risdate><volume>26</volume><issue>5</issue><spage>1090</spage><epage>1123</epage><pages>1090-1123</pages><issn>0036-1429</issn><eissn>1095-7170</eissn><coden>SJNAEQ</coden><abstract>This paper analyzes multigrid preconditioners designed to accelerate the convergence of various singularly perturbed boundary value problems for which the coefficients of low-order derivative terms are multiplied by a large parameter K. The multigrid preconditioner is not applied to the given operator, but rather to a simpler operator (such as the Laplacian) for which its implementation can be carried out much more efficiently. 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| ispartof | SIAM journal on numerical analysis, 1989-10, Vol.26 (5), p.1090-1123 |
| issn | 0036-1429 1095-7170 |
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| source | SIAM Journals; JSTOR Mathematics & Statistics; JSTOR Archive Collection A-Z Listing |
| subjects | Applied mathematics Boundary value problems Eigenvalues Eigenvectors Exact sciences and technology Fourier analysis Initial guess Integers Iterative methods Mathematics Methods Multigrid methods Numerical analysis Numerical analysis. Scientific computation Partial differential equations, boundary value problems Preconditioning Sciences and techniques of general use |
| title | Analysis and Application of Multigrid Preconditioners for Singularly Perturbed Boundary Value Problems |
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