A novel equi-dimensional finite element method for flow and transport in fractured porous media satisfying discrete maximum principle and conservation properties
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Numerical simulations of flow and transport in porous media usually rely on hybrid-dimensional models, i.e., the fracture is considered as objects of a lower dimension compared to the embedding matrix. Such models are usually combined with non-conforming discretizations as they avoid the inherent difficulties associated with the generation of meshes that explicitly resolve fractures-matrix interfaces. However, non-conforming discretizations demand a more complicated coupling of different sub-models and may require special care to ensure conservative fluxes. We propose a novel approach for the simulation of flow and transport problems in fractured porous media based on an equi-dimensional representation of the fractures. The major challenge for these types of representation is the creation of meshes which resolve the several complex interfaces between the fractures and the embedding matrix. To overcome this difficulty, we employ a strategy based on adaptive mesh refinement (AMR). The idea at the base of the proposed AMR is to start from an initially uniform coarse mesh and refine the elements which have non-empty overlaps with at least one of the fractures. Iterating this process allows to create non-uniform non-conforming meshes, which do not resolve the interfaces but can approximate them with arbitrary accuracy. We demonstrate that low-order finite element (FE) discretizations on adapted meshes are globally and locally conservative and we suitably adapt an algebraic flux correction technique to ensure the discrete maximum principle. In particular, we show that the notorious conditions on M-matrices have to be adapted to the basis functions defined on non-conforming meshes. Although the proposed applications come from geophysical applications, the obtained results could be applied to any diffusion and transport problems, on both conforming and non-conforming meshes.
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