The Nonconforming Virtual Element Method for a Stationary Stokes Hemivariational Inequality with Slip Boundary Condition
- PDF / 418,870 Bytes
- 19 Pages / 439.37 x 666.142 pts Page_size
- 97 Downloads / 173 Views
The Nonconforming Virtual Element Method for a Stationary Stokes Hemivariational Inequality with Slip Boundary Condition Min Ling1 · Fei Wang2
· Weimin Han1,3
Received: 10 March 2020 / Revised: 23 August 2020 / Accepted: 8 October 2020 © Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract In this paper, the nonconforming virtual element method is studied to solve a hemivariational inequality problem for the stationary Stokes equations with a nonlinear slip boundary condition. The nonconforming virtual elements enriched with polynomials on slip boundary are used to discretize the velocity, and discontinuous piecewise polynomials are used to approximate the pressure. The inf-sup condition is shown for the nonconforming virtual element method. An error estimate is derived under appropriate solution regularity assumptions, and the error bound is of optimal order when lowest-order virtual elements for the velocity and piecewise constants for the pressure are used. A numerical example is presented to illustrate the theoretically predicted convergence order. Keywords Virtual element method · Hemivariational inequality · Stokes problem · Slip boundary condition
The work of this author was partially supported by the National Natural Science Foundation of China (Grant No. 11771350).
B
Fei Wang [email protected] Min Ling [email protected] Weimin Han [email protected]
1
School of Mathematics and Statistics, Xi’an Jiaotong University, Xi’an 710049, Shaanxi, China
2
School of Mathematics and Statistics and State Key Laboratory of Multiphase Flow in Power Engineering, Xi’an Jiaotong University, Xi’an 710049, Shaanxi, China
3
Department of Mathematics and Program in Applied Mathematical and Computational Sciences, University of Iowa, Iowa City, IA 52242, USA 0123456789().: V,-vol
123
56
Page 2 of 19
Journal of Scientific Computing
(2020) 85:56
1 Introduction Hemivariational inequality (HVI), concerning nonsmooth and nonconvex functionals, represents a powerful tool in the study of a large number of nonlinear boundary value problems. Mathematical theory, numerical approximations and applications of hemivariational inequalities can be found in several comprehensive References [11,20,24,26,27]. Recently, optimal order error estimates are derived for the linear finite element solutions of hemivariational inequalities, see [18,19] for a summary account. In [30], the interior penalty discontinuous Galerkin method is studied for solving an elliptic hemivariational inequality for semipermeable media, and optimal convergence order is proved for the linear element. The conforming virtual element method (VEM) for a second-order elliptic problem was initially introduced in [3] as a generalization of the classical finite element method to accommodate arbitrary element-geometry. The nonconforming VEM for the same problem was constructed later in [2], where the corresponding virtual element can be viewed as an extension of the Crouzeix–Raviart element to general polygonal meshes. Because
Data Loading...
