The two-grid decoupled method for hybrid-dimensional fracture models based on the discontinuous Galerkin method

In this paper, we consider a kind of hybrid-dimensional fracture models, in which fractures are treated as $(n-1)$-dimensional objects in a n-dimensional porous medium, and the interaction between fractures and surrounding domain is taken into account. For the model with one single fracture, the discontinuous Galerkin (DG) method is firstly proposed, and the optimal order error estimate in the discrete $H^1$-norm for the pressure is derived. Moreover, a two-grid decoupled method is considered, which uses a coarse grid rough approximations of the interface variables to decouple the mixed domain problem on a fine grid, and the optimal error estimate is also analyzed. The proposed DG method and two-grid decoupled method are then extended to a model with complex intersecting fractures, as well as the corresponding theoretical analysis. Numerical examples with both one single fracture and intersecting fractures are all presented to verity the accuracy of the considered methods.