A New Projection-Based Stabilized Virtual Element Method for the Stokes Problem
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A New Projection-Based Stabilized Virtual Element Method for the Stokes Problem Jun Guo1 · Minfu Feng1 Received: 15 March 2020 / Revised: 25 July 2020 / Accepted: 2 September 2020 © Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract We propose and analyze a stabilized virtual element method for the Stokes problem on polytopal meshes. We employ the C 0 continuous arbitrary “equal-order” virtual element pairs to approximate both velocity and pressure, and develop a projection-based stabilization term to circumvent the discrete inf-sup condition, then we obtain the corresponding error estimates. The presented method involves neither the projection of the second derivative nor additional coupling terms, and it is parameter-free. In particularly, for the lowest-order case on triangular (tetrahedral) meshes the stabilized method introduced by Bochev et al. (SIAM J. Numer. Anal. 44: 82–101, 2006) is a special case of our method up to an approximation of the load term. Furthermore, numerical results are shown to confirm the theoretical predictions. Keywords Virtual element method · Stokes problem · Projection · Inf-sup condition
1 Introduction The incompressible flow originates from many physical problems, such as shale gas exploration, groundwater sewage treatment and tidal power generation [27]. Due to the practical requirements of such problems, various methods that allow for polygonal and polyhedral meshes have been proposed anddeveloped in the past few years. An incomplete list of them is: the polygonal finite element method [38], the mimetic finite difference method (MFD) [19], the local discontinuous Galerkin method [40], the hybridizable discontinuous Galerkin method [31,37], the weak Galerkin finite element method [36], the hybrid high-order method [9], the virtual element method (VEM) [12,24,34], and so on. Recently, as a further development of the combination of MFD and classical finite element method (FEM), VEM technology has drawn extensive concerns [11,20,26]. It not only bears to attack these problems with arbitrary regularity, but also supports general polytopal meshes especially including the distorted and non-convex elements. Because of its remarkable fea-
Foundation item: Supported by the National Natural Science Foundation of China (No. 11971337 ) and the Key Fund Project of Sichuan Provincial Department of Education (No. 18ZA0276).
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Minfu Feng [email protected] College of Mathematics, Sichuan University, Chengdu 610064, China 0123456789().: V,-vol
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tures, VEM technology has been employed to solve various problems, such as second-order elliptic problems [23], parabolic problems [1], hyperbolic problems [41], eigenvalue problems [35], elasticity problems [21,39], and non-linear problems [5], for instance. Besides, there has been many crucial research for Stokes and Navier-Stokes problem in the mixed VEM framework. Specifically, one can refer to divergence free H 1 -conforming virtual elements [24,25], H (
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