Mixed and nonconforming finite element methods on a system of polygons

We investigate the lowest-order Raviart–Thomas mixed finite element method for second-order elliptic problems posed over a system of intersecting two-dimensional polygons placed in three-dimensional Euclidean space. Such problems arise for example in the context of groundwater flow through granitoid massifs, where the polygons represent the rock fractures. The domain is characteristic by the presence of intersection lines shared by three or more polygons. We first construct continuous and discrete function spaces ensuring the continuity of scalar functions and an appropriate continuity of the normal trace of vector functions across such intersection lines. We then propose a variant of the lowest-order Raviart–Thomas mixed finite element method for the given problem with the domain discretized into a triangular mesh and prove its well-posedness. We finally investigate the relation of the hybridization of the considered mixed finite element method to the piecewise linear nonconforming finite element method. We extend the results known in this direction onto networks of polygons, general diffusion tensors, and general boundary conditions. The obtained relation enables in particular an efficient implementation of the mixed finite element method. We finally verify the theoretical results on a model problem with a known analytical solution and show the application of the proposed method to the simulation of a real problem.