Penalty method with P1/P1 finite element approximation for the Stokes equations under the slip boundary condition

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• 作者：Takahito Kashiwabara ; Issei Oikawa ; Guanyu Zhou
• 关键词：Mathematics Subject Classification65N30 ; 35Q30
• 刊名：Numerische Mathematik
• 出版年：2016
• 出版时间：December 2016
• 年：2016
• 卷：134
• 期：4
• 页码：705-740
• 全文大小：1,639 KB
• 刊物类别：Mathematics and Statistics
• 刊物主题：Mathematics
  Numerical Analysis
  Mathematics
  Mathematical and Computational Physics
  Mathematical Methods in Physics
  Numerical and Computational Methods
  Applied Mathematics and Computational Methods of Engineering
  
• 出版者：Springer Berlin / Heidelberg
• ISSN：0945-3245
• 卷排序：134

文摘

We consider the P1/P1 or P1b/P1 finite element approximations to the Stokes equations in a bounded smooth domain subject to the slip boundary condition. A penalty method is applied to address the essential boundary condition \(u\cdot n = g\) on \(\partial \Omega \), which avoids a variational crime and simultaneously facilitates the numerical implementation. We give \(O(h^{1/2} + \epsilon ^{1/2} + h/\epsilon ^{1/2})\)-error estimate for velocity and pressure in the energy norm, where h and \(\epsilon \) denote the discretization parameter and the penalty parameter, respectively. In the two-dimensional case, it is improved to \(O(h + \epsilon ^{1/2} + h^2/\epsilon ^{1/2})\) by applying reduced-order numerical integration to the penalty term. The theoretical results are confirmed by numerical experiments.