A lowest-order locking-free nonconforming virtual element method based on the reduced integration technique for linear elasticity problems

We develop a lowest-order nonconforming virtual element method for planar linear elasticity, which can be viewed as an extension of the idea in Falk (1991) to the virtual element method (VEM), with the family of polygonal meshes satisfying a very general geometric assumption. The method is shown to be uniformly convergent for the nearly incompressible case with optimal rates of convergence. The crucial step is to establish the discrete Korn's inequality, yielding the coercivity of the discrete bilinear form. We also provide a unified locking-free scheme both for the conforming and nonconforming VEMs in the lowest-order case. Numerical results validate the feasibility and effectiveness of the proposed numerical algorithms.

Automated summary

We propose a lowest-order nonconforming virtual element method for planar linear elasticity, extending the idea in Falk (1991) to the virtual element method (VEM), with a general geometric assumption for the family of polygonal meshes. The method is shown to have uniform convergence and optimal rates of convergence in the nearly incompressible case. The establishment of the discrete Korn's inequality is a crucial step for achieving the coercivity of the discrete bilinear form. We also provide a locking-free scheme for both conforming and nonconforming VEMs in the lowest-order case. Numerical results validate the feasibility and effectiveness of the proposed numerical algorithms.