Jessika Camaño, Cristian Muñoz, Ricardo Oyarzua:
Analysis of a mixed finite element method for the Poisson problem with data in L^p, 2n/(n + 2) < p < 2, n = 2, 3
In this paper we analyze the numerical approximation of the Poisson problem in mixed form, considering a right-hand side f \in L^p(\Omega), with p \in (2n/(n+2),2), where n = 2,3 is the dimension of \Omega. The analysis of the corresponding continuous and discrete problems are carried out by means of the classical Babuska-Brezzi theory, where the associated Galerkin scheme is defined by Raviart-Thomas elements of lowest order combined with piecewise constants. In particular, we prove well-posedness and convergence of the discrete scheme under a quasi-uniformity condition of the mesh. Next, we apply the theory developed for the Poisson problem to a convection-difussion problem, providing well-posedness of the continuous and discrete problems and optimal convergence. Finally, we corroborate the theoretical results with suitable numerical results in two and three dimensions.