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2018•Communications in Computational PhysicsRequires access

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

Jiachuan Zhang, Kai Zhang, Jingzhi Li, Xiaoshen Wang

Open publisher page 40 citations

Abstract

In this paper, a weak Galerkin finite element method (WGFEM) is proposed for solving the Navier-Stokes equations (NSEs). The existence and uniqueness of the WGFEM solution of NSEs are established. The WGFEM provides very accurate numerical approximations for both the velocity field and pressure field, even with very high Reynolds numbers. The salient feature is that the flexibility of the WGFEM for the choice of the order of the velocity and pressure comparing to the standard finite element methods. Optimal order error estimates in both |·|$W_h$ and $L^2$ norms are proved for the semi-discretized scheme. Numerical simulations are presented to show the efficiency of the WGFEM.

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What this paper is about

In this paper, a weak Galerkin finite element method (WGFEM) is proposed for solving the Navier-Stokes equations (NSEs). The existence and uniqueness of the WGFEM solution of NSEs are established. The WGFEM provides very accurate numerical approximations for both the velocity field and pressure field, even with very high Reynolds numbers. The salient feature is that the flexibility of the WGFEM for the choice of the order of the velocity and pressure comparing to the standard finite element methods. Optimal order error estimates in both |·|$W_h$ and $L^2$ norms are proved for the semi-discretized scheme. Numerical simulations are presented to show the efficiency of the WGFEM.

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Available abstract

In this paper, a weak Galerkin finite element method (WGFEM) is proposed for solving the Navier-Stokes equations (NSEs). The existence and uniqueness of the WGFEM solution of NSEs are established. The WGFEM provides very accurate numerical approximations for both the velocity field and pressure field, even with very high Reynolds numbers. The salient feature is that the flexibility of the WGFEM for the choice of the order of the velocity and pressure comparing to the standard finite element methods. Optimal order error estimates in both |·|$W_h$ and $L^2$ norms are proved for the semi-discretized scheme. Numerical simulations are presented to show the efficiency of the WGFEM.

Key concepts: Finite element method, Galerkin method, Discontinuous Galerkin method, Navier–Stokes equations, Extended finite element method, Mathematics, Mixed finite element method, Applied mathematics

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