The nonconforming virtual element method for the Navier-Stokes equations
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The nonconforming virtual element method for the Navier-Stokes equations Xin Liu1,2 · Zhangxin Chen1,3,4
Received: 23 May 2017 / Accepted: 14 March 2018 © Springer Science+Business Media, LLC, part of Springer Nature 2018
Abstract In this paper a unified nonconforming virtual element scheme for the Navier-Stokes equations with different dimensions and different polynomial degrees is described. Its key feature is the treatment of general elements including non-convex and degenerate elements. According to the properties of an enhanced nonconforming virtual element space, the stability of this scheme is proved based on the choice of a proper velocity and pressure pair. Furthermore, we establish optimal error estimates in the discrete energy norm for velocity and the L2 norm for both velocity and pressure. Finally, we test some numerical examples to validate the theoretical results. Keywords Nonconforming virtual element method · Navier-Stokes equations · General elements · Stability · Energy and L2 optimal error estimates
Communicated by: Ilaria Perugia Research is supported in part by Foundation CMG in Xi’an Jiaotong University and the scholarship from China Scholarship Council (CSC) under the Grant CSC No.201706280335 Zhangxin Chen
[email protected] Xin Liu [email protected] 1
School of Mathematics and Statistics, Xi’an Jiaotong University, Xi’an, 710049, China
2
Istituto di Matematica Applicata e Tecnologie Informatiche del C.N.R, via Ferrata 5a, 27100 Pavia, Italy
3
College of Petroleum Engineering, China University of Petroleum, Beijing, 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, Z. Chen
Mathematics Subject Classification (2010) 65N30 · 65N12 · 65N15 · 76D05
1 Introduction The steady flow of a viscous incompressible fluid in a polygonal or polyhedral domain ⊆ Rd (d = 2, 3) is described by the Navier-Stokes equations: ⎧ in , ⎨ −νu + (u · ∇)u + ∇p = f div u = 0 in , (1.1) ⎩ u=0 on ∂, where u, p, f, and ν, respectively, represent the velocity, pressure, external body force, and positive constant viscosity. Our aim is to construct and analyze the nonconforming virtual element method (VEM) for problem (1.1) in its diffusion-dominated regime. We first present a brief overview of the nonconforming finite element method. This method enforces the continuity of its basis functions at k Gauss-Legendre points on edges or faces, which ensures the optimal convergence rate by the minimal continuity requirement [1]. But the different behaviors of odd and even k-order polynomial approximations [1–7] and the shape of elements (including elements in different dimensions) [8–12] make the construction of nonconforming finite element schemes different; i.e., it is difficult to give a unified scheme for odd and even k, and different elements and dimensions simultaneously. The conforming VEM has been proposed [14] and implemented [28] for the Poisson equation. In addition, t
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