High order discontinuous Galerkin method for reduced flow models in fractured porous media

We construct a new symmetric interior penalty discontinuous Galerkin (SIPDG)-method for incompressible flow in fractured porous media. The method is developed for computing accurate approximations of the reduced flow model, where fractures are treated as lower dimensional objects. Unlike previous formulations, the methodology proposed herein shows ability to capture the asymptotic limits of highly permeable and fracture seals, also covering a wider range of values of the quadrature integration parameter which appears in the pressure jumps across the fractures. As a first novel contribution we should mention that the method was developed for specific interface condition in the reduced problem for which known in literature DG method are not applicable. As a second novelty we propose a new penalty technique for stabilization of the method and obtain explicit estimate for the penalty parameters associated with flow in matrix and fractures in order to achieve stability. And finally we derive new high order hp type a priori error estimates for the numerical solution in the energy norm. Numerical results illustrate the performance of the proposed SIPDG-method in simulating discrete fracture models.