The Finite element method (FEM) is a natural numerical discretization approach for variational inequalities. The classical FEM works on the elements with simple geometries, like triangles and rectangles. Due to the flexibility on constructing the local function space, the discontinuous Galerkin (DG) method can handle very general meshes with hanging nodes, which make them very suitable for hp-adaptivity. However, the DG method needs large number of degrees of freedom. Recently, borrowing the ideas from the mimetic finite difference method, the virtual element method  (VEM), which can be regarded as a generalization of the classic FEM to general polygonal meshes, have been developed for solving a variety of partial differential equation. The virtual element method can handle very general (non-convex) polygonal elements with an arbitrary number of edges. In addition, it allows geometrical hanging nodes in the meshes, even they are treated as vertices of the polygonal elements in practice, so it is easy to implement h-adaptive strategy for the VEM. We study the virtual element method for solving obstacle problem [10], simplified friction problem [11], and plate frictional contact problem [12]. The optimal error estimates are established for the lowest-order virtual element methods, and numerical examples are given to support the theoretical results.  

[10] Fei Wang and Huayi Wei, Virtual Element Methods for Obstacle Problem,  IMA Journal of Numerical Analysis, 40 (2020), 708–728, (published online 2018).

[11] Fei Wang* and Huayi Wei, Virtual Element Method for Simplified Friction Problem, Applied Mathematics Letters, (2018) https://doi.org/10.1016/j.aml.2018.06.002.

[12] Fei Wang and Jikun Zhao, Conforming and Nonconforming Virtual Element Methods for a Kirchhoff Plate Contact Problem, IMA Journal of Numerical Analysis 41 (2021) , 1496–1521.

[13] Yanling Deng, Fei Wang* and Huayi Wei, A Posteriori Error Estimates of Virtual Element Method for a Simplified Friction Problem, Journal of Scientific Computing, 83:52 (2020), DOI: 10.1007/s10915-020-01242-9. 

[14] Min Ling, Fei Wang* and Weimin Han, The Nonconforming Virtual Element Method for a Stationary Stokes Hemivariational Inequality with Slip Boundary Condition, Journal of Scientific Computing, (2020) 85:56，https://doi.org/10.1007/s10915-020-01333-7. 

[15] Fei Wang*, Bangmin Wu and Weimin Han, The Virtual Element Method for General Elliptic Hemivariational Inequalities, Journal of Computational and Applied Mathematics, 389 (2021) 113330.