A new stabilized finite volume method for the stationary Stokes equations
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A new stabilized finite volume method for the stationary Stokes equations Jian Li · Zhangxin Chen
Received: 18 September 2006 / Accepted: 1 June 2007 / Published online: 22 November 2007 © Springer Science + Business Media, LLC 2007
Abstract In this paper we develop and study a new stabilized finite volume method for the two-dimensional Stokes equations. This method is based on a local Gauss integration technique and the conforming elements of the lowest-equal order pair (i.e., the P1 − P1 pair). After a relationship between this method and a stabilized finite element method is established, an error estimate of optimal order in the H 1 norm for velocity and an estimate in the L2 -norm for pressure are obtained. An optimal error estimate in the L2 -norm for the velocity is derived under an additional assumption on the body force. Keywords Stokes equations · Finite element method · Finite volume method · Inf-sup condition · Error estimate Mathematics Subject Classifications (2000) 35Q10 · 65N30 · 76D05
Communicated by: Jinchao Xu. This work is supported in part by the NSF of China 10701001 and by the US National Science Foundation grant DMS-0609995 and CMG Chair Funds in Reservoir Simulation. J. Li Department of Mathematics, Baoji University of Arts and Sciences, Baoji 721007, People’s Republic of China e-mail: [email protected] Z. Chen Department of Chemical and Petroleum Engineering, Schulich School of Engineering, University of Calgary, 2500 University Drive N.W. Calgary, Alberta T2N 1N4, Canada J. Li · Z. Chen (B) Faculty of Science, Xi’an Jiaotong University, Xi’an 710049, People’s Republic of China e-mail: [email protected]
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J. Li, Z. Chen
1 Introduction Finite difference, finite element, and finite volume methods are three major numerical tools for solving partial differential equations. Among these methods, the finite volume method is the most intuitive because it is based on local conservation of mass, momentum, or energy over volumes (control volumes or co-volumes). This method is a numerical technique that lies somewhere between the finite element and finite difference methods; it has a flexibility similar to that of the finite element method for handling complicated geometries, and its implementation is comparable to that of the finite difference method. Moreover, its numerical solutions usually have certain conservation features that are desirable in many practical applications. The finite volume method is also termed the control volume method, the covolume method, or the first-order generalized difference method. Many papers were devoted to its error analysis for second-order elliptic and parabolic partial differential problems [9, 12, 13, 16, 30, 36]. Error estimates of optimal order in the H 1 -norm are the same as those for the linear finite element method [12, 20, 29, 31]. Error estimates of optimal order in the L2 -norm can be obtained as well [13, 20]. The finite volume method for generalized Stokes problems was studied by many people [14, 15, 17, 32, 37]. They analyzed this method by us
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