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Volume 15, Issue 1-2
The Singularity-Separated Method for the Singular Perturbation Problems in 1-D

Chuanmiao Chen & Jing Yang

Int. J. Numer. Anal. Mod., 15 (2018), pp. 102-110.

Published online: 2018-01

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  • Abstract

The singularity-separated method (SSM) for the singular perturbation problem $-\epsilon u''+bu' + cu = f(x), u(0) = u(1) = 0$, is proposed for the first time. The solution is expressed as $u = w-ν$, where $w$ is the solution of corresponding third boundary value problem and $ν$ is an exact singular function. We have proved a global regularity, $||w||_2 ≤ C$, where the constant $C$ is independent of $\epsilon$, and discussed three kinds of finite element (FE) methods with SSM. Numerical results show that these FE-solutions have the high accuracy when only one element in boundary layer is taken.

  • Keywords

Singular perturbation problem, singularity-separated method, third boundary value, second order regularity, finite elements.

  • AMS Subject Headings

34E10, 34E15, 65N30

  • Copyright

COPYRIGHT: © Global Science Press

  • Email address

cmchen@hunnu.edu.cn (Chuanmiao Chen)

jingyang@hunau.edu.cn (Jing Yang)

  • BibTex
  • RIS
  • TXT
@Article{IJNAM-15-102, author = {Chen , Chuanmiao and Yang , Jing}, title = {The Singularity-Separated Method for the Singular Perturbation Problems in 1-D}, journal = {International Journal of Numerical Analysis and Modeling}, year = {2018}, volume = {15}, number = {1-2}, pages = {102--110}, abstract = {

The singularity-separated method (SSM) for the singular perturbation problem $-\epsilon u''+bu' + cu = f(x), u(0) = u(1) = 0$, is proposed for the first time. The solution is expressed as $u = w-ν$, where $w$ is the solution of corresponding third boundary value problem and $ν$ is an exact singular function. We have proved a global regularity, $||w||_2 ≤ C$, where the constant $C$ is independent of $\epsilon$, and discussed three kinds of finite element (FE) methods with SSM. Numerical results show that these FE-solutions have the high accuracy when only one element in boundary layer is taken.

}, issn = {2617-8710}, doi = {https://doi.org/}, url = {http://global-sci.org/intro/article_detail/ijnam/10558.html} }
TY - JOUR T1 - The Singularity-Separated Method for the Singular Perturbation Problems in 1-D AU - Chen , Chuanmiao AU - Yang , Jing JO - International Journal of Numerical Analysis and Modeling VL - 1-2 SP - 102 EP - 110 PY - 2018 DA - 2018/01 SN - 15 DO - http://doi.org/ UR - https://global-sci.org/intro/article_detail/ijnam/10558.html KW - Singular perturbation problem, singularity-separated method, third boundary value, second order regularity, finite elements. AB -

The singularity-separated method (SSM) for the singular perturbation problem $-\epsilon u''+bu' + cu = f(x), u(0) = u(1) = 0$, is proposed for the first time. The solution is expressed as $u = w-ν$, where $w$ is the solution of corresponding third boundary value problem and $ν$ is an exact singular function. We have proved a global regularity, $||w||_2 ≤ C$, where the constant $C$ is independent of $\epsilon$, and discussed three kinds of finite element (FE) methods with SSM. Numerical results show that these FE-solutions have the high accuracy when only one element in boundary layer is taken.

Chen , Chuanmiao and Yang , Jing. (2018). The Singularity-Separated Method for the Singular Perturbation Problems in 1-D. International Journal of Numerical Analysis and Modeling. 15 (1-2). 102-110. doi:
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