A Difference Virtual Element Method for the 3D Elliptic Equation with the Variable Coefficient on General Cylindrical Domains

In this paper, we present and analysis a difference virtual element method (DVEM) for the three dimensional (3D) elliptic equation on general cylindrical domains. This method combines the dimension splitting method and operator splitting technique to transform the virtual element solution of 3D elliptic equation into a series of virtual element solution of 2D elliptic equation based on $(x, y)$ plane, where the central difference discretization is adopted in the $z$-direction. This allows us to solve partial differential equations on cylindrical domains at the low cost in mesh generation compared with 3D virtual element method. The $H^1$-norm error estimation of the DVEM is analysed in this paper. Finally, some numerical examples are performed to verify the theoretical predictions and showcase the efficiency of the proposed method.