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首页> 外文期刊>Computer Methods in Applied Mechanics and Engineering >A priori and a posteriori error analyses of a velocity-pseudostress formulation for a class of quasi-Newtonian Stokes flows
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A priori and a posteriori error analyses of a velocity-pseudostress formulation for a class of quasi-Newtonian Stokes flows

机译：一类拟牛顿斯托克斯流的速度-拟应力公式的先验和后验误差分析

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摘要

In this paper we introduce and analyze new mixed finite element schemes for a class of nonlinear Stokes models arising in quasi-Newtonian fluids. The methods are based on a non-standard mixed approach in which the velocity, the pressure, and the pseudostress are the original unknowns. However, we use the incompressibility condition to eliminate the pressure, and set the velocity gradient as an auxiliary unknown, which yields a twofold saddle point operator equation as the resulting dual-mixed variational formulation. In addition, a suitable augmented version of the latter showing a single saddle point structure is also considered. We apply known results from nonlinear functional analysis to prove that the corresponding continuous and discrete schemes are well-posed. In particular, we show that Raviart-Thomas elements of order k≥0 for the pseudostress, and piecewise polynomials of degree k for the velocity and its gradient, ensure the well-posedness of the associated Galerkin schemes. Moreover, we prove that any finite element subspace of the square integrable tensors can be employed to approximate the velocity gradient in the case of the augmented formulation. Then, we derive a reliable and efficient residual-based a posteriori error estimator for each scheme. Finally, we provide several numerical results illustrating the good performance of the resulting mixed finite element methods, confirming the theoretical properties of the estimator, and showing the behaviour of the associated adaptive algorithms.
机译：在本文中，我们介绍和分析了拟牛顿流体中产生的一类非线性Stokes模型的新混合有限元方案。这些方法基于非标准混合方法，其中速度，压力和拟应力是原始未知数。但是，我们使用不可压缩条件消除压力，并将速度梯度设置为辅助未知数，这将生成一个双重鞍点算子方程式作为最终的双混合变分公式。另外，还考虑了后者的合适的增强形式，其示出了单个鞍形点结构。我们应用非线性功能分析的已知结果来证明相应的连续和离散方案是正确的。特别地，我们表明伪应力的k≥0阶的Raviart-Thomas元素和速度及其梯度的k阶分段多项式确保了相关Galerkin方案的适定性。此外，我们证明了在可扩展公式的情况下，平方可积张量的任何有限元子空间都可用于近似速度梯度。然后，我们为每种方案导出了可靠且有效的基于残差的后验误差估计器。最后，我们提供了一些数值结果，这些结果说明了所得混合有限元方法的良好性能，确认了估计器的理论性质，并显示了相关自适应算法的行为。

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