Mixed Virtual Element Methods For Nearly Incompressible Linear Elasticity Equations And Reissner-Mindlin Plate Problem

Two kind of mixed virtual element methods are proposed for numerically solving the nearly incompressible linear elasticity problem with a Lame coefficient and ReissnerMindlin plate problem with a thicdness t respectively.A new virtual element method is proposed for numerically solving the nearly incompressible linear elasticity problem which involves a Lame coefficient which would lead to the locking phenomenon as λ→∞.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.To establish the Babuska-Brezzi inf-sup condition,two kinds of stabilizations are designed.The stability and the error estimates are proven uniformly in the Lame 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 Lame coefficient,and the new method is lockingfree for λ→∞.Numerical results are presented to illustrate the performance and the theoretical results of the new method.A new virtual element method on arbitrary polygonal meshes is proposed for the Reissner-Mindlin plate model,where the families of the Poisson-like virtual elements of any equal-order or unequal-order virtual elements[Wk]2-Wm are used for approximating the rotation and the transverse placement while the shear stress is element locally computed by local L2 projections from the discontinuous polynomial elements[Pl]2 for m-1≤l≤min(m+1,k).Stability and error estimates are established,which hold for all values of the plate thickness(including the zero thickness)and for all variables(rotation,transverse displacement and shear stress)in their norms independent of the mesh size and the plate thickness.Numerical experiments are performed to illustrate the method and the theoretical results.