Volume 33, Issue 3
Convergence Analysis for the Iterated Defect Correction Scheme of Finite Element Methods on Rectangle Grids

Youai Li

J. Comp. Math., 33 (2015), pp. 297-306.

Published online: 2015-06

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  • Abstract

This paper develops a new method to analyze convergence of the iterated defect correction scheme of finite element methods on rectangular grids in both two and three dimensions. The main idea is to formulate energy inner products and energy (semi)norms into matrix forms. Then, two constants of two key inequalities involved are min and max eigenvalues of two associated generalized eigenvalue problems, respectively. Local versions on the element level of these two generalized eigenvalue problems are exactly solved to obtain sharp (lower) upper bounds of these two constants. This and some essential observations for iterated solutions establish convergence in 2D and the monotone decreasing property in 3D. For two dimensions the results herein improve those in literature; for three dimensions the results herein are new. Numerical results are presented to examine theoretical results.

  • Keywords

Petrov-Galerkin method, iterated defect correction scheme, convergence, eigenvalue problem.

  • AMS Subject Headings

65N30, 65N15, 35J25.

  • Copyright

COPYRIGHT: © Global Science Press

  • Email address

liya@th.btbu.edu.cn (Youai Li)

  • BibTex
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  • TXT
@Article{JCM-33-297, author = {Li , Youai}, title = {Convergence Analysis for the Iterated Defect Correction Scheme of Finite Element Methods on Rectangle Grids}, journal = {Journal of Computational Mathematics}, year = {2015}, volume = {33}, number = {3}, pages = {297--306}, abstract = {

This paper develops a new method to analyze convergence of the iterated defect correction scheme of finite element methods on rectangular grids in both two and three dimensions. The main idea is to formulate energy inner products and energy (semi)norms into matrix forms. Then, two constants of two key inequalities involved are min and max eigenvalues of two associated generalized eigenvalue problems, respectively. Local versions on the element level of these two generalized eigenvalue problems are exactly solved to obtain sharp (lower) upper bounds of these two constants. This and some essential observations for iterated solutions establish convergence in 2D and the monotone decreasing property in 3D. For two dimensions the results herein improve those in literature; for three dimensions the results herein are new. Numerical results are presented to examine theoretical results.

}, issn = {1991-7139}, doi = {https://doi.org/10.4208/jcm.1501-m4426}, url = {http://global-sci.org/intro/article_detail/jcm/9843.html} }
TY - JOUR T1 - Convergence Analysis for the Iterated Defect Correction Scheme of Finite Element Methods on Rectangle Grids AU - Li , Youai JO - Journal of Computational Mathematics VL - 3 SP - 297 EP - 306 PY - 2015 DA - 2015/06 SN - 33 DO - http://doi.org/10.4208/jcm.1501-m4426 UR - https://global-sci.org/intro/article_detail/jcm/9843.html KW - Petrov-Galerkin method, iterated defect correction scheme, convergence, eigenvalue problem. AB -

This paper develops a new method to analyze convergence of the iterated defect correction scheme of finite element methods on rectangular grids in both two and three dimensions. The main idea is to formulate energy inner products and energy (semi)norms into matrix forms. Then, two constants of two key inequalities involved are min and max eigenvalues of two associated generalized eigenvalue problems, respectively. Local versions on the element level of these two generalized eigenvalue problems are exactly solved to obtain sharp (lower) upper bounds of these two constants. This and some essential observations for iterated solutions establish convergence in 2D and the monotone decreasing property in 3D. For two dimensions the results herein improve those in literature; for three dimensions the results herein are new. Numerical results are presented to examine theoretical results.

Youai Li. (2020). Convergence Analysis for the Iterated Defect Correction Scheme of Finite Element Methods on Rectangle Grids. Journal of Computational Mathematics. 33 (3). 297-306. doi:10.4208/jcm.1501-m4426
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