Adaptive interior penalty hybridized discontinuous Galerkin methods for Darcy flow in fractured porous media

Abstract

Abstract In this paper, we design and analyze an interior penalty hybridized discontinuous Galerkin (IP-HDG) method for the Darcy flow in the two- and three-dimensional fractured porous media. The discrete fracture model is used to model the fractures. The piecewise polynomials of degree $k$ are employed to approximate the pressure in the fractures and the pressure in the surrounding porous media. We prove that the IP-HDG method is well posed if the penalty parameter is large enough. Based on the discrete solutions of pressures, the discrete Darcy velocity in the matrix and the reduced fractures can be recovered, respectively, to be locally mass-conservative. A robust residual-based a posteriori error estimator is established for an energy-norm of pressure. Finally, numerical results are provided to show the efficiency of the proposed a posteriori error estimator.