A mass conserving mixed stress formulation for the Stokes equations
A mass conserving mixed stress formulation for the Stokes equations
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DOI：
10.1093/imanum/drz022
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发表时间：
2020-07-01
影响因子：
2.1
通讯作者：
Schoeberl, Joachim
中科院分区：
文献类型：
--
作者：
Gopalakrishnan, Jay;Lederer, Philip L.;Schoeberl, Joachim
我们提出了Stokes方程的应力公式。速度u近似与H（div）一致的有限元提供精确的质量守恒。虽然许多标准方法使用H-1-协调空间的离散速度H（div）-符合考虑变分公式在这项工作中。在新的函数空间H（curl div）内设置等于速度梯度的新的类似应力的变量σ。新的矩阵值有限元具有连续的“法向切向”分量构造H（旋度div）的近似函数。误差分析的结论是u中的误差（以离散H-1范数测量）、sigma中的误差（以L-2测量）和压力p（也以L-2测量）的最佳收敛速率。精确的质量守恒性质与另一个称为压力鲁棒性的结构保持性质直接相关，如压力无关的速度误差估计所示。的计算成本衡量的接口自由度是可比的旧的和新的斯托克斯离散。
We propose stress formulation of the Stokes equations. The velocity u is approximated with H(div)-conforming finite elements providing exact mass conservation. While many standard methods use H-1-conforming spaces for the discrete velocity H(div)-conformity fits the considered variational formulation in this work. A new stress-like variable sigma equalling the gradient of the velocity is set within a new function space H(curl div). New matrix-valued finite elements having continuous 'normal-tangential' components are constructed to approximate functions in H(curl div). An error analysis concludes with optimal rates of convergence for errors in u (measured in a discrete H-1-norm), errors in sigma (measured in L-2) and the pressure p (also measured in L-2). The exact mass conservation property is directly related to another structure-preservation property called pressure robustness, as shown by pressure-independent velocity error estimates. The computational cost measured in terms of interface degrees of freedom is comparable to old and new Stokes discretizations.