A Weak Galerkin Finite Element Method for the Navier-Stokes Equations

Zhang, Jiachuan; Zhang, Kai; Li, Jingzhi; Wang, Xiaoshen

doi:10.4208/cicp.oa-2016-0267

Cited by 28 publications

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“…On the other hand, in the past two decades many researchers have also developed discontinuous Galerkin methods and weak Galerkin methods for approximating incompressible Navier-Stokes equations with high Reynolds numbers effectively (cf. [7,8,[21][22][23]). Therefore, it is very valuable for us to propose and analyze the related generalized Arrow-Hurwicz methods for the preceding discrete problems.…”

Section: Discussionmentioning

confidence: 99%

The Generalized Arrow-Hurwicz Method with Applications to Fluid Computation

Du¹,

Huang²

2019

CICP

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In this paper, we first discuss the existence and uniqueness of a class of nonlinear saddle-point problems, which are frequently encountered in physical models. Then, a generalized Arrow-Hurwicz method is introduced to solve such problems. For the method, the convergence rate analysis is established under some reasonable conditions. It is also applied to solve three typical discrete methods in fluid computation, with the computational efficiency demonstrated by a series of numerical experiments.

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Section: Discussionmentioning

confidence: 99%

The Generalized Arrow-Hurwicz Method with Applications to Fluid Computation

Du¹,

Huang²

2019

CICP

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“…Using the Lemma A.1 (trace inequality) in [30] and referring the proof of Lemma 3.3 in [37], we can get the following inequality estimates.…”

Section: Lemma 22 (Cf Lemma 41 Inmentioning

confidence: 99%

“…This makes the computations of the WGFEM and that of the conforming finite element are comparable. The WGFEM has been developed for various PDEs and we refer to Stokes' equations [22,31,28], the Brinkman equations [20,15,32], stochastic jump coefficients problem [18], Maxwell's equations [19], the Navier-Stokes Equations [17,37], and references therein.…”

Section: Introductionmentioning

confidence: 99%

An adaptive weak Galerkin finite element method with hierarchical bases for the elliptic problem

Zhang

et al. 2020

Numerical Methods Partial

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Based on a posteriori error estimator with hierarchical bases, an adaptive weak Galerkin finite element method (WGFEM) is proposed for the elliptic problem with mixed boundary conditions. For the posteriori error estimator, we are only required to solve a linear algebraic system with diagonal entries corresponding to the degree of freedoms, which significantly reduces the computational cost. The upper and lower bounds of the error estimator are shown to addresses the reliability and efficiency of the adaptive approach. Numerical simulations are provided to demonstrate the effectiveness and robustness of the proposed method.

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“…Experiments with the GrFEM have shown an increase in economy and in the nodal accuracy compared to FE solutions of the Burgers' equations [3,5,6] and for other problems [8,4,9,27,32] . Recently, the weak Galerkin finite element method has catched much consideration in the field of numericaln PDEs , The idea of the weak Galerkin method was first introduced by Wang and Ye [14], the method was applied to the second order elliptic equations [1,15,16,19,20,22,24,28], the Stokes equations [17,21], Parabolic equations [10,11,25], biharmonic equations [23,26], Navier-Stokes equations [7,18,30] and 1-D Burgers' equation [31], etc. In this paper we present (WGFEM) and (WGrFEM) for 2-D Burgers' problem with a fully-discrete approximation for the time variable, the backward-difference formula for the time variable is examined and the stability and error estimate are proved for these methods.…”

Section: Introductionmentioning

confidence: 99%

Weak Galerkin and Weak Group Finite Element Methods for 2-D Burgers' Problem

Noon¹

2020

ARFMTS

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In this paper, we apply weak Galerkin finite element method (WGFEM) and weak group finite element method (WGrFEM) for 2-D Burgers' problem by using the weak functions and their corresponding discrete weak derivatives in standard weak form. We consider the fully formulation using the backward-difference form for the time variable and we prove the stability and an error estimate for these methods. The numerical examples of our methods have been carried out through implementation in MATLAB programs and compared with the exact solutions and other available literature to illustrate the efficiency of the proposed methods, the WGFEM and the WGrFEM provide convergent approximations and handles the equation well in different cases, the results obtained of our methods are very acceptable and competent more than the results available in the other literature. Moreover, a WGrFEM is displayed to be better than the GFEM.

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A Weak Galerkin Finite Element Method for the Navier-Stokes Equations

Cited by 28 publications

References 0 publications

The Generalized Arrow-Hurwicz Method with Applications to Fluid Computation

The Generalized Arrow-Hurwicz Method with Applications to Fluid Computation

An adaptive weak Galerkin finite element method with hierarchical bases for the elliptic problem

Weak Galerkin and Weak Group Finite Element Methods for 2-D Burgers' Problem

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