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Volume 42, Issue 5
A Uniform Convergent Petrov-Galerkin Method for a Class of Turning Point Problems

Li Feng & Zhongyi Huang

DOI: 10.4208/jcm.2305-m2022-0171

J. Comp. Math., 42 (2024), pp. 1356-1379.

Published online: 2024-07

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  • Abstract

In this paper, we propose a numerical method for turning point problems in one dimension based on Petrov-Galerkin finite element method (PGFEM). We first give a priori estimate for the turning point problem with a single boundary turning point. Then we use PGFEM to solve it, where test functions are the solutions to piecewise approximate dual problems. We prove that our method has a first-order convergence rate in both $L^∞_h$ norm and a discrete energy norm when we select the exact solutions to dual problems as test functions. Numerical results show that our scheme is efficient for turning point problems with different types of singularities, and the convergency coincides with our theoretical results.

  • Keywords

Turning point problem, Petrov-Galerkin finite element method, Uniform convergency.

  • AMS Subject Headings

65N06, 65B99

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{JCM-42-1356, author = {Feng , Li and Huang , Zhongyi}, title = {A Uniform Convergent Petrov-Galerkin Method for a Class of Turning Point Problems}, journal = {Journal of Computational Mathematics}, year = {2024}, volume = {42}, number = {5}, pages = {1356--1379}, abstract = {

In this paper, we propose a numerical method for turning point problems in one dimension based on Petrov-Galerkin finite element method (PGFEM). We first give a priori estimate for the turning point problem with a single boundary turning point. Then we use PGFEM to solve it, where test functions are the solutions to piecewise approximate dual problems. We prove that our method has a first-order convergence rate in both $L^∞_h$ norm and a discrete energy norm when we select the exact solutions to dual problems as test functions. Numerical results show that our scheme is efficient for turning point problems with different types of singularities, and the convergency coincides with our theoretical results.

}, issn = {1991-7139}, doi = {https://doi.org/10.4208/jcm.2305-m2022-0171}, url = {http://global-sci.org/intro/article_detail/jcm/23281.html} }
TY - JOUR T1 - A Uniform Convergent Petrov-Galerkin Method for a Class of Turning Point Problems AU - Feng , Li AU - Huang , Zhongyi JO - Journal of Computational Mathematics VL - 5 SP - 1356 EP - 1379 PY - 2024 DA - 2024/07 SN - 42 DO - http://doi.org/10.4208/jcm.2305-m2022-0171 UR - https://global-sci.org/intro/article_detail/jcm/23281.html KW - Turning point problem, Petrov-Galerkin finite element method, Uniform convergency. AB -

In this paper, we propose a numerical method for turning point problems in one dimension based on Petrov-Galerkin finite element method (PGFEM). We first give a priori estimate for the turning point problem with a single boundary turning point. Then we use PGFEM to solve it, where test functions are the solutions to piecewise approximate dual problems. We prove that our method has a first-order convergence rate in both $L^∞_h$ norm and a discrete energy norm when we select the exact solutions to dual problems as test functions. Numerical results show that our scheme is efficient for turning point problems with different types of singularities, and the convergency coincides with our theoretical results.

Feng , Li and Huang , Zhongyi. (2024). A Uniform Convergent Petrov-Galerkin Method for a Class of Turning Point Problems. Journal of Computational Mathematics. 42 (5). 1356-1379. doi:10.4208/jcm.2305-m2022-0171
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