$$H(\textrm{div})$$ -conforming HDG methods for the stress-velocity formulation of the Stokes equations and the Navier–Stokes equations

IF 2.1 2区 数学 Q1 MATHEMATICS, APPLIED
Weifeng Qiu, Lina Zhao
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Abstract

In this paper we devise and analyze a pressure-robust and superconvergent HDG method in stress-velocity formulation for the Stokes equations and the Navier–Stokes equations with strongly symmetric stress. The stress and velocity are approximated using piecewise polynomial space of order k and \(H(\textrm{div};\Omega )\)-conforming space of order \(k+1\), respectively, where k is the polynomial order. In contrast, the tangential trace of the velocity is approximated using piecewise polynomials of order k. Moreover, the characterization of the proposed schemes shows that the globally coupled unknowns are the normal trace and the tangential trace of velocity, and the piecewise constant approximation for the trace of the stress. The discrete \(H^1\)-stability is established for the discrete solution. The proposed formulation yields divergence-free velocity, but causes difficulties for the derivation of the pressure-independent error estimate given that the pressure variable is not employed explicitly in the discrete formulation. This difficulty can be overcome by observing that the \(L^2\) projection to the stress space has a nice commuting property. Moreover, superconvergence for velocity in discrete \(H^1\)-norm is obtained, with regard to the degrees of freedom of the globally coupled unknowns. Then the convergence of the discrete solution to the weak solution for the Navier–Stokes equations via the compactness argument is rigorously analyzed under minimal regularity assumption. The strong convergence for velocity and stress is proved. Importantly, the strong convergence for velocity in discrete \(H^1\)-norm is achieved. Several numerical experiments are carried out to confirm the proposed theories.

Abstract Image

用于斯托克斯方程和纳维-斯托克斯方程的应力-速度公式的 $$H(\textrm{div})$$ -conforming HDG 方法
在本文中,我们设计并分析了一种压力稳健、超收敛的HDG方法,该方法采用应力-速度公式计算斯托克斯方程和具有强对称应力的纳维-斯托克斯方程。应力和速度分别使用k阶的分次多项式空间和\(k+1\)阶的(H(\textrm{div};\Omega )\)符合空间近似,其中k是多项式阶数。此外,所提方案的特征表明,全局耦合未知量是速度的法线迹和切线迹,以及应力迹的片断常数近似值。离散解建立了离散(H^1)稳定性。所提出的公式可以得到无发散的速度,但由于离散公式中没有明确使用压力变量,这给推导与压力无关的误差估计带来了困难。这一困难可以通过观察应力空间的 \(L^2\) 投影具有良好的换向特性来克服。此外,就全局耦合未知数的自由度而言,在离散 \(H^1\)-norm 中获得了速度的超收敛性。然后,在最小正则性假设下,通过紧凑性论证严格分析了纳维-斯托克斯方程的离散解对弱解的收敛性。证明了速度和应力的强收敛性。重要的是,实现了速度在离散(H^1\)规范下的强收敛性。为了证实所提出的理论,还进行了一些数值实验。
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来源期刊
Numerische Mathematik
Numerische Mathematik 数学-应用数学
CiteScore
4.10
自引率
4.80%
发文量
72
审稿时长
6-12 weeks
期刊介绍: Numerische Mathematik publishes papers of the very highest quality presenting significantly new and important developments in all areas of Numerical Analysis. "Numerical Analysis" is here understood in its most general sense, as that part of Mathematics that covers: 1. The conception and mathematical analysis of efficient numerical schemes actually used on computers (the "core" of Numerical Analysis) 2. Optimization and Control Theory 3. Mathematical Modeling 4. The mathematical aspects of Scientific Computing
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