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Volume 26, Issue 3
On the $P_1$ Powell-Sabin Divergence-Free Finite Element for the Stokes Equations

Shangyou Zhang

J. Comp. Math., 26 (2008), pp. 456-470.

Published online: 2008-06

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  • Abstract

The stability of the $P_1$-$P_0$ mixed-element   is established on general Powell-Sabin triangular grids.  The piecewise linear finite element solution approximating   the velocity is divergence-free pointwise   for the Stokes equations.  The finite element solution approximating the pressure in   the Stokes equations can be obtained as a byproduct if   an iterative method is adopted for solving the discrete   linear system of equations.  Numerical tests are presented confirming the theory on the stability and the optimal order of convergence for the $P_1$ Powell-Sabin   divergence-free finite element method.

  • Keywords

Powell Sabin triangles, Mixed finite elements, Stokes, Divergence-free element.

  • AMS Subject Headings

65M60, 65N30, 76D07.

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{JCM-26-456, author = {Shangyou Zhang}, title = {On the $P_1$ Powell-Sabin Divergence-Free Finite Element for the Stokes Equations}, journal = {Journal of Computational Mathematics}, year = {2008}, volume = {26}, number = {3}, pages = {456--470}, abstract = {

The stability of the $P_1$-$P_0$ mixed-element   is established on general Powell-Sabin triangular grids.  The piecewise linear finite element solution approximating   the velocity is divergence-free pointwise   for the Stokes equations.  The finite element solution approximating the pressure in   the Stokes equations can be obtained as a byproduct if   an iterative method is adopted for solving the discrete   linear system of equations.  Numerical tests are presented confirming the theory on the stability and the optimal order of convergence for the $P_1$ Powell-Sabin   divergence-free finite element method.

}, issn = {1991-7139}, doi = {https://doi.org/}, url = {http://global-sci.org/intro/article_detail/jcm/8636.html} }
TY - JOUR T1 - On the $P_1$ Powell-Sabin Divergence-Free Finite Element for the Stokes Equations AU - Shangyou Zhang JO - Journal of Computational Mathematics VL - 3 SP - 456 EP - 470 PY - 2008 DA - 2008/06 SN - 26 DO - http://doi.org/ UR - https://global-sci.org/intro/article_detail/jcm/8636.html KW - Powell Sabin triangles, Mixed finite elements, Stokes, Divergence-free element. AB -

The stability of the $P_1$-$P_0$ mixed-element   is established on general Powell-Sabin triangular grids.  The piecewise linear finite element solution approximating   the velocity is divergence-free pointwise   for the Stokes equations.  The finite element solution approximating the pressure in   the Stokes equations can be obtained as a byproduct if   an iterative method is adopted for solving the discrete   linear system of equations.  Numerical tests are presented confirming the theory on the stability and the optimal order of convergence for the $P_1$ Powell-Sabin   divergence-free finite element method.

Shangyou Zhang. (2008). On the $P_1$ Powell-Sabin Divergence-Free Finite Element for the Stokes Equations. Journal of Computational Mathematics. 26 (3). 456-470. doi:
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