Convergence analysis of a fast virtual element method for coupled Darcy flows in fractured porous media on general polygonal meshes

In this article, we propose a novel two-grid discretization for the approximation of coupled Darcy flows in fractured porous media on general polygonal meshes. The Finite Element Method (FEM) is used to discreting the reduced fracture equation, and the Virtual Element Method (VEM) is applied to the matrix equation. The core of our fast VEM lies in the utilization of two meshes with distinct grid sizes. On the coarse grid, we solve the original coupled problem, and subsequently, we employ the coarse grid solution to decouple the equations on the finer grid. Optimal error estimates in $L^2$ and semi-$H^1$ norm for the VEM solution and semi-$H^1$ norm for the fast VEM solution are rigorously provided, respectively, which shows our algorithm is capable of achieving asymptotically optimal estimation while significantly minimizing computational expenses.