A Posteriori Error Estimation for an Interior Penalty Type Method Employing $H(\mathrm{div})$ Elements for the Stokes Equations
A Posteriori Error Estimation for an Interior Penalty Type Method Employing $H(\mathrm{div})$ Elements for the Stokes Equations
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DOI：
10.1137/100783996
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发表时间：
2011-02
期刊：
影响因子：
--
通讯作者：
Junping Wang;Yanqi Wang;X. Ye
中科院分区：
文献类型：
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作者：
Junping Wang;Yanqi Wang;X. Ye
本文建立了用$H（\mathrm{div}）$有限元内点罚型方法离散Stokes方程的后验误差分析。然后采用后验误差估计器设计两种网格细化策略，一种是局部的，另一种是全局的。基于局部的细化技术被认为是能够捕捉局部奇异的数值解。Stokes问题的数值公式使用Raviart-Thomas型的$H（\mathrm{div}）$协调元。因此，有限元解的特点是完全满足连续性方程（质量守恒）。本文的结果为该方法的可靠性和有效性提供了严格的分析。特别地，得到了一个H^1 $范数后验误差估计，以及上界和下界估计。数值结果验证了后验误差估计的新理论。
This paper establishes a posteriori error analysis for the Stokes equations discretized by an interior penalty type method using $H(\mathrm{div})$ finite elements. The a posteriori error estimator is then employed for designing two grid refinement strategies; one is locally based and the other is globally based. The locally based refinement technique is believed to be able to capture local singularities in the numerical solution. The numerical formulations for the Stokes problem make use of $H(\mathrm{div})$ conforming elements of Raviart-Thomas type. Therefore, the finite element solution features a full satisfaction of the continuity equation (mass conservation). The result of this paper provides a rigorous analysis for the method's reliability and efficiency. In particular, an $H^1$ norm a posteriori error estimator is obtained, together with upper and lower bound estimates. Numerical results are presented to verify the new theory of a posteriori error estimators.