Analysis of an HDG method for the Navier-Stokes equations with Dirac measures

IF 1.9 3区 数学 Q2 Mathematics
Haitao Leng
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引用次数: 0

Abstract

In two dimensions, we analyze a hybridized discontinuous Galerkin (HDG) method for the Navier-Stokes equations with Dirac measures. The approximate velocity field obtained by the HDG method is shown to be pointwise divergence-free and $H$(div)-conforming. Under a smallness assumption on the continuous and discrete solutions, a posteriori error estimator, that provides an upper bound for the $L^2$-norm in the velocity, is proposed in the convex domain. In the polygonal domain, reliable and efficient a posteriori error estimator for the $W^{1,q}$-seminorm in the velocity and $L^q$-norm in the pressure is also proved. Finally, a Banach's fixed point iteration method and an adaptive HDG algorithm are introduced to solve the discrete system and show the performance of the obtained a posteriori error estimators.
具有Dirac测度的Navier-Stokes方程的HDG方法分析
在二维空间中,我们分析了具有Dirac测度的Navier-Stokes方程的一种杂化不连续伽辽金方法。用HDG方法得到的近似速度场是无点发散的,符合$H$(div)。在对连续解和离散解的小假设下,在凸域上提出了一个后验误差估计量,该估计量为速度上的L^2 -范数提供了上界。在多边形域,证明了速度上的$W^{1,q}$-半模和压力上的$L^q$-范数的可靠和有效的后验误差估计。最后,介绍了Banach不动点迭代法和自适应HDG算法对离散系统进行求解,并展示了得到的后验误差估计的性能。
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来源期刊
CiteScore
2.70
自引率
5.30%
发文量
27
审稿时长
6-12 weeks
期刊介绍: M2AN publishes original research papers of high scientific quality in two areas: Mathematical Modelling, and Numerical Analysis. Mathematical Modelling comprises the development and study of a mathematical formulation of a problem. Numerical Analysis comprises the formulation and study of a numerical approximation or solution approach to a mathematically formulated problem. Papers should be of interest to researchers and practitioners that value both rigorous theoretical analysis and solid evidence of computational relevance.
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