Jessika Camaño, Cristian Muñoz, Ricardo Oyarzúa:
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.
This preprint gave rise to the following definitive publication(s):
Jessika CAMAñO, Cristian MUñOZ, Ricardo OYARZúA: Numerical analysis of a dual-mixed problem in non-standard Banach spaces. Electronic Transactions on Numerical Analysis, vol. 48, pp. 114-130, (2018).