Volume 12, Issue 3
A Finite Element Dual Singular Function Method to Solve the Stokes Equations Including Corner Singularities

Int. J. Numer. Anal. Mod., 12 (2015), pp. 516-535.

Published online: 2015-12

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• Abstract

The finite element dual singular function method [FE-DSFM] has been constructed and analyzed accuracy by Z. Cai and S. Kim to solve the Laplace equation on a polygonal domain with one reentrant corner. In this paper, we impose FE-DSFM to solve the Stokes equations via the mixed finite element method. To do this, we compute the singular and the dual singular functions analytically at a non-convex corner. We prove well-posedness by using the contraction mapping theorem and then estimate errors of the algorithm. We obtain optimal accuracy $O(h)$ for velocity in $\rm{H}^1(Ω)$ and pressure in $L^2(Ω)$, but we are able to prove only $O(h^{1+\lambda})$ error bounds for velocity in $\rm{L}^2(\Omega)$ and stress intensity factor, where $\lambda$ is the eigenvalue (solution of (4)). However, we get optimal accuracy results in numerical experiments.

65M12, 65M15, 76D05

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@Article{IJNAM-12-516, author = {}, title = {A Finite Element Dual Singular Function Method to Solve the Stokes Equations Including Corner Singularities}, journal = {International Journal of Numerical Analysis and Modeling}, year = {2015}, volume = {12}, number = {3}, pages = {516--535}, abstract = {

The finite element dual singular function method [FE-DSFM] has been constructed and analyzed accuracy by Z. Cai and S. Kim to solve the Laplace equation on a polygonal domain with one reentrant corner. In this paper, we impose FE-DSFM to solve the Stokes equations via the mixed finite element method. To do this, we compute the singular and the dual singular functions analytically at a non-convex corner. We prove well-posedness by using the contraction mapping theorem and then estimate errors of the algorithm. We obtain optimal accuracy $O(h)$ for velocity in $\rm{H}^1(Ω)$ and pressure in $L^2(Ω)$, but we are able to prove only $O(h^{1+\lambda})$ error bounds for velocity in $\rm{L}^2(\Omega)$ and stress intensity factor, where $\lambda$ is the eigenvalue (solution of (4)). However, we get optimal accuracy results in numerical experiments.

}, issn = {2617-8710}, doi = {https://doi.org/}, url = {http://global-sci.org/intro/article_detail/ijnam/500.html} }
TY - JOUR T1 - A Finite Element Dual Singular Function Method to Solve the Stokes Equations Including Corner Singularities JO - International Journal of Numerical Analysis and Modeling VL - 3 SP - 516 EP - 535 PY - 2015 DA - 2015/12 SN - 12 DO - http://doi.org/ UR - https://global-sci.org/intro/article_detail/ijnam/500.html KW - Stokes equations, dual singular function method, corner singularity, incompressible fluids. AB -

The finite element dual singular function method [FE-DSFM] has been constructed and analyzed accuracy by Z. Cai and S. Kim to solve the Laplace equation on a polygonal domain with one reentrant corner. In this paper, we impose FE-DSFM to solve the Stokes equations via the mixed finite element method. To do this, we compute the singular and the dual singular functions analytically at a non-convex corner. We prove well-posedness by using the contraction mapping theorem and then estimate errors of the algorithm. We obtain optimal accuracy $O(h)$ for velocity in $\rm{H}^1(Ω)$ and pressure in $L^2(Ω)$, but we are able to prove only $O(h^{1+\lambda})$ error bounds for velocity in $\rm{L}^2(\Omega)$ and stress intensity factor, where $\lambda$ is the eigenvalue (solution of (4)). However, we get optimal accuracy results in numerical experiments.

Jae-Hong Pyo. (1970). A Finite Element Dual Singular Function Method to Solve the Stokes Equations Including Corner Singularities. International Journal of Numerical Analysis and Modeling. 12 (3). 516-535. doi:
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