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A weak Galerkin finite element method for the Oseen equations Xin Liu1 · Jian Li2,3 · Zhangxin Chen1,4

Received: 5 November 2015 / Accepted: 23 June 2016 © Springer Science+Business Media New York 2016

Abstract In this paper, a weak Galerkin finite element method for the Oseen equations of incompressible fluid flow is proposed and investigated. This method is based on weak gradient and divergence operators which are designed for the finite element discontinuous functions. Moreover, by choosing the usual polynomials of degree i ≥ 1 for the velocity and polynomials of degree i − 1 for the pressure and enhancing the polynomials of degree i − 1 on the interface of a finite element partition for the velocity, this new method has a lot of attractive computational features: more general finite element partitions of arbitrary polygons or polyhedra with certain shape regularity, fewer degrees of freedom and parameter free. Stability and error estimates of optimal order are obtained by defining a weak convection term. Finally, a series

Communicated by: A. Zhou Research supported in part by NSF of China (No. 11371031), and the Key Projects of Baoji university of Arts and Sciences (No. ZK15040) and (No. ZK15033).  Zhangxin Chen

[email protected] Xin Liu [email protected] Jian Li [email protected] 1

School of Mathematics and Statistics, Xi’an Jiaotong University, Xi’an 710049, China

2

Department of Mathematics, Baoji University of Arts and Sciences, Baoji 721013, China

3

Department of Mathematics, Shaanxi University of Sciences and Technology, Xi’an, 710021, China

4

Department of Chemical and Petroleum Engineering, Schulich School of Engineering, University of Calgary, 2500 University Drive N.W., Calgary, Alberta T2N 1N4, Canada

X. Liu et al.

of numerical experiments are given to show that this method has good stability and accuracy for the Oseen problem. Keywords Weak Galerkin · Finite element method · The Oseen equations · More general partitions Mathematics Subject Classification (2010) 65N30 · 65N12 · 65N15 · 76D07

1 Introduction In this paper, we propose a weak Galerkin finite element method for Oseen equations. As an extension of the standard finite elements, the weak Galerkin method substitutes the classical operators (e.g., gradient, divergence, and curl) by weakly defined operators according to integration by parts. The idea of the weak Galerkin method has been introduced and analyzed in [1] for second-order elliptic problems based on local RT or BDM elements, which limited a finite element partition to triangles or tetrahedra. Then, in [4], the weak Galerkin method was extended to allow arbitrary shapes of finite elements in a partition by applying a stabilization idea, which provides a convenient flexibility in mesh generation. A computational process for the weak Galerkin method for second-order elliptic equations with more general finite element partitions has been explained in [5]. In [6], the possibility of an optimal combination of polynomial spaces which minimizes the number of unknowns has