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Volume 16, Issue 2
The Defect Iteration of the Finite Element for Elliptic Boundary Value Problems and Petrov-Galerkin Approximation

Junbin Gao, Yidu Yang & T. M. Shih

J. Comp. Math., 16 (1998), pp. 152-164.

Published online: 1998-04

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  • Abstract

In this paper we introduce a Petrov-Galerkin approximation model to the solution of linear and semi-linear elliptic boundary value problems in which piecewise quadratic polynomial space and piecewise linear polynomial space are used as the shape function space and the test function space, respectively. We prove that the approximation order of the standard quadratic finite element can be attained in this Petrov-Galerkin model. Based on the so-called "contractivity" of the interpolation operator, we further prove that the defect iterative sequence of the linear finite element solution converge to the proposed Petrov-Galerkin approximate solution.

  • Keywords

Petrov-Galerkin approximation, defect iteration correction, interpolation operator.

  • AMS Subject Headings

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COPYRIGHT: © Global Science Press

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@Article{JCM-16-152, author = {Gao , JunbinYang , Yidu and Shih , T. M.}, title = {The Defect Iteration of the Finite Element for Elliptic Boundary Value Problems and Petrov-Galerkin Approximation}, journal = {Journal of Computational Mathematics}, year = {1998}, volume = {16}, number = {2}, pages = {152--164}, abstract = {

In this paper we introduce a Petrov-Galerkin approximation model to the solution of linear and semi-linear elliptic boundary value problems in which piecewise quadratic polynomial space and piecewise linear polynomial space are used as the shape function space and the test function space, respectively. We prove that the approximation order of the standard quadratic finite element can be attained in this Petrov-Galerkin model. Based on the so-called "contractivity" of the interpolation operator, we further prove that the defect iterative sequence of the linear finite element solution converge to the proposed Petrov-Galerkin approximate solution.

}, issn = {1991-7139}, doi = {https://doi.org/}, url = {http://global-sci.org/intro/article_detail/jcm/9149.html} }
TY - JOUR T1 - The Defect Iteration of the Finite Element for Elliptic Boundary Value Problems and Petrov-Galerkin Approximation AU - Gao , Junbin AU - Yang , Yidu AU - Shih , T. M. JO - Journal of Computational Mathematics VL - 2 SP - 152 EP - 164 PY - 1998 DA - 1998/04 SN - 16 DO - http://doi.org/ UR - https://global-sci.org/intro/article_detail/jcm/9149.html KW - Petrov-Galerkin approximation, defect iteration correction, interpolation operator. AB -

In this paper we introduce a Petrov-Galerkin approximation model to the solution of linear and semi-linear elliptic boundary value problems in which piecewise quadratic polynomial space and piecewise linear polynomial space are used as the shape function space and the test function space, respectively. We prove that the approximation order of the standard quadratic finite element can be attained in this Petrov-Galerkin model. Based on the so-called "contractivity" of the interpolation operator, we further prove that the defect iterative sequence of the linear finite element solution converge to the proposed Petrov-Galerkin approximate solution.

Gao , JunbinYang , Yidu and Shih , T. M.. (1998). The Defect Iteration of the Finite Element for Elliptic Boundary Value Problems and Petrov-Galerkin Approximation. Journal of Computational Mathematics. 16 (2). 152-164. doi:
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