Volume 14, Issue 4-5
Finite Element Method to Control the Domain Singularities of Poisson Equation Using the Stress Intensity Factor: Mixed Boundary Condition

Int. J. Numer. Anal. Mod., 14 (2017), pp. 500-510.

Published online: 2017-08

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In this article, we consider the Poisson equation on a polygonal domain with the domain singularity raised from the changed boundary conditions with the inner angle $\omega>\frac{\pi}{2}$. The solution of the Poisson equation with such singularity has a singular decomposition: regular part plus singular part. The singular part is a linear combination of one or two singular functions. The coefficients of the singular functions are usually called stress intensity factors and can be computed by the extraction formula. In  we introduced a new partial differential equation which has 'zero' stress intensity factor using this stress intensity factor, from whose solution we can obtain a very accurate solution of the original problem simply by adding singular part. Although the method in  works well for the Poisson problem with Dirichlet boundary condition, it does not give optimal results for the case with stronger singularity, for example, mixed boundary condition with bigger inner angle. In this paper we give a revised algorithm which gives optimal convergences for both cases.

65F10, 65N30

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@Article{IJNAM-14-500, author = {}, title = {Finite Element Method to Control the Domain Singularities of Poisson Equation Using the Stress Intensity Factor: Mixed Boundary Condition}, journal = {International Journal of Numerical Analysis and Modeling}, year = {2017}, volume = {14}, number = {4-5}, pages = {500--510}, abstract = {

In this article, we consider the Poisson equation on a polygonal domain with the domain singularity raised from the changed boundary conditions with the inner angle $\omega>\frac{\pi}{2}$. The solution of the Poisson equation with such singularity has a singular decomposition: regular part plus singular part. The singular part is a linear combination of one or two singular functions. The coefficients of the singular functions are usually called stress intensity factors and can be computed by the extraction formula. In  we introduced a new partial differential equation which has 'zero' stress intensity factor using this stress intensity factor, from whose solution we can obtain a very accurate solution of the original problem simply by adding singular part. Although the method in  works well for the Poisson problem with Dirichlet boundary condition, it does not give optimal results for the case with stronger singularity, for example, mixed boundary condition with bigger inner angle. In this paper we give a revised algorithm which gives optimal convergences for both cases.

}, issn = {2617-8710}, doi = {https://doi.org/}, url = {http://global-sci.org/intro/article_detail/ijnam/10046.html} }
TY - JOUR T1 - Finite Element Method to Control the Domain Singularities of Poisson Equation Using the Stress Intensity Factor: Mixed Boundary Condition JO - International Journal of Numerical Analysis and Modeling VL - 4-5 SP - 500 EP - 510 PY - 2017 DA - 2017/08 SN - 14 DO - http://doi.org/ UR - https://global-sci.org/intro/article_detail/ijnam/10046.html KW - Finite element, singular function, dual singular function, stress intensity factor. AB -

In this article, we consider the Poisson equation on a polygonal domain with the domain singularity raised from the changed boundary conditions with the inner angle $\omega>\frac{\pi}{2}$. The solution of the Poisson equation with such singularity has a singular decomposition: regular part plus singular part. The singular part is a linear combination of one or two singular functions. The coefficients of the singular functions are usually called stress intensity factors and can be computed by the extraction formula. In  we introduced a new partial differential equation which has 'zero' stress intensity factor using this stress intensity factor, from whose solution we can obtain a very accurate solution of the original problem simply by adding singular part. Although the method in  works well for the Poisson problem with Dirichlet boundary condition, it does not give optimal results for the case with stronger singularity, for example, mixed boundary condition with bigger inner angle. In this paper we give a revised algorithm which gives optimal convergences for both cases.

Seokchan Kim & Hyung-Chun Lee. (1970). Finite Element Method to Control the Domain Singularities of Poisson Equation Using the Stress Intensity Factor: Mixed Boundary Condition. International Journal of Numerical Analysis and Modeling. 14 (4-5). 500-510. doi:
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