Optimal convergence analysis of the virtual element methods for second-order Sobolev equations with variable coefficients on polygonal meshes

In case of variable coefficients, the discrete bilinear forms in the virtual element discretization do not satisfy the consistency and stability properties. Also, for distinct damping and diffusion coefficient, it is difficult to obtain optimal convergence rate in \(L^{2}\) $L^2$ norm. We discuss the virtual element method for the linear Sobolev equations with variable coefficients. The consistency and stability estimates for the discrete bilinear forms are derived. For optimal convergence analysis a new non-standard projection operator is introduced. We have used implicit second order Newmark scheme for the fully discrete approximation. Numerical experiments are illustrated to confirm our theoretical findings.