The Stabilized Nonconforming Virtual Element Method for Linear Elasticity Problem

摘要

Abstract We present the stabilized nonconforming virtual element method for linear elasticity problem in two dimensions. The jump penalty term is introduced to guarantee the stability of the discrete formulation as the stabilization term, which is obtained based on the discrete Korn’s inequality. In order to obtain the computability of jump penalty term, we reconstruct the lowest-order nonconforming virtual element by imposing some restrictions on the conforming virtual element space of order 2. We prove the interpolation error estimate for the virtual element and the ellipticity of the discrete bilinear form, so the resulting stabilized method is well-posed. Then we show the optimal convergence in the L2documentclass12pt{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$$L^2$$end{document} and H1documentclass12pt{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$$H^1$$end{document} norms. Moreover, this method is locking-free, i.e. the convergence is uniform with respect to the Lamé constant. Numerical results are provided to confirm the theoretical results.