Convergence Analysis of the Preconditioned Group Splitting Methods in Boundary Value Problems
The construction of a specific splitting-type preconditioner in block formulation applied to a class of group relaxation iterative methods derived from the centred and rotated (skewed) finite difference approximations has been shown to improve the convergence rates of these methods. In this paper, w...
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| Format: | Article |
| Language: | English |
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Wiley
2012-01-01
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| Series: | Abstract and Applied Analysis |
| Online Access: | http://dx.doi.org/10.1155/2012/867598 |
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| _version_ | 1832553735355629568 |
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| author | Norhashidah Hj. Mohd Ali Abdulkafi Mohammed Saeed |
| author_facet | Norhashidah Hj. Mohd Ali Abdulkafi Mohammed Saeed |
| author_sort | Norhashidah Hj. Mohd Ali |
| collection | DOAJ |
| description | The construction of a specific splitting-type preconditioner in block formulation applied to a class of group relaxation iterative methods derived from the centred and rotated (skewed) finite difference approximations has been shown to improve the convergence rates of these methods. In this paper, we present some theoretical convergence analysis on this preconditioner specifically applied to the linear systems resulted from these group iterative schemes in solving an elliptic boundary value problem. We will theoretically show the relationship between the spectral radiuses of the iteration matrices of the preconditioned methods which affects the rate of convergence of these methods. We will also show that the spectral radius of the preconditioned matrices is smaller than that of their unpreconditioned counterparts if the relaxation parameter is in a certain optimum range. Numerical experiments will also be presented to confirm the agreement between the theoretical and the experimental results. |
| format | Article |
| id | doaj-art-2d83900483ea490182bc79510f05f1ad |
| institution | Kabale University |
| issn | 1085-3375 1687-0409 |
| language | English |
| publishDate | 2012-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | Abstract and Applied Analysis |
| spelling | doaj-art-2d83900483ea490182bc79510f05f1ad2025-02-03T05:53:18ZengWileyAbstract and Applied Analysis1085-33751687-04092012-01-01201210.1155/2012/867598867598Convergence Analysis of the Preconditioned Group Splitting Methods in Boundary Value ProblemsNorhashidah Hj. Mohd Ali0Abdulkafi Mohammed Saeed1School of Mathematical Sciences, Universitiy Sains Malaysia, 11800 USM, Pulau Pinang, MalaysiaSchool of Mathematical Sciences, Universitiy Sains Malaysia, 11800 USM, Pulau Pinang, MalaysiaThe construction of a specific splitting-type preconditioner in block formulation applied to a class of group relaxation iterative methods derived from the centred and rotated (skewed) finite difference approximations has been shown to improve the convergence rates of these methods. In this paper, we present some theoretical convergence analysis on this preconditioner specifically applied to the linear systems resulted from these group iterative schemes in solving an elliptic boundary value problem. We will theoretically show the relationship between the spectral radiuses of the iteration matrices of the preconditioned methods which affects the rate of convergence of these methods. We will also show that the spectral radius of the preconditioned matrices is smaller than that of their unpreconditioned counterparts if the relaxation parameter is in a certain optimum range. Numerical experiments will also be presented to confirm the agreement between the theoretical and the experimental results.http://dx.doi.org/10.1155/2012/867598 |
| spellingShingle | Norhashidah Hj. Mohd Ali Abdulkafi Mohammed Saeed Convergence Analysis of the Preconditioned Group Splitting Methods in Boundary Value Problems Abstract and Applied Analysis |
| title | Convergence Analysis of the Preconditioned Group Splitting Methods in Boundary Value Problems |
| title_full | Convergence Analysis of the Preconditioned Group Splitting Methods in Boundary Value Problems |
| title_fullStr | Convergence Analysis of the Preconditioned Group Splitting Methods in Boundary Value Problems |
| title_full_unstemmed | Convergence Analysis of the Preconditioned Group Splitting Methods in Boundary Value Problems |
| title_short | Convergence Analysis of the Preconditioned Group Splitting Methods in Boundary Value Problems |
| title_sort | convergence analysis of the preconditioned group splitting methods in boundary value problems |
| url | http://dx.doi.org/10.1155/2012/867598 |
| work_keys_str_mv | AT norhashidahhjmohdali convergenceanalysisofthepreconditionedgroupsplittingmethodsinboundaryvalueproblems AT abdulkafimohammedsaeed convergenceanalysisofthepreconditionedgroupsplittingmethodsinboundaryvalueproblems |