A mixed virtual element method for nearly incompressible linear elasticity equations

A new virtual element method is proposed for numerically solving the nearly incompressible linear elasticity problem which involves a Lamé coefficient λ which would lead to the locking phenomenon as λ→∞\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\lambda \rightarrow \infty $\end{document}. We use the classical mixed formulation in terms of the displacement and the multiplier. In the new method, both displacement and multiplier are approximated by the any equal-order or any unequal-order virtual element spaces which are generated from the scalar Laplace operator −Δ. To establish the Babus̆ka-Brezzi inf-sup condition, two kinds of stabilizations are designed. The stability and the error estimates are proven uniformly in the Lamé coefficient, where the optimal error estimates in H1-norm and L2-norm are obtained for the displacement and the corresponding error bounds are also obtained for the multiplier. The error bounds obtained are uniform in the Lamé coefficient, and the new method is locking-free for λ→∞\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\lambda \rightarrow \infty $\end{document}. Numerical results are presented to illustrate the performance and the theoretical results of the new method.