Jessika Camaño, Luis F. Gatica, Ricardo Oyarzúa:
A priori and a posteriori error analyses of a flux-based mixed-FEM for convection-diffusion-reaction problems
In this paper we propose and analyze a new mixed-type finite element method for the numerical simulation of a dffusion-convection-reaction problem with non-homogeneous Dirichlet boundary condition. The method is based on a new formulation of the problem of interest consisting in a single variational equation posed in H(div;\Omega), where the flux (or gradient of the primal variable u) is the main and only unknown. Consequently, we propose a conforming Raviart-Thomas approximation of order k \ge 0 for the flux, and the primal unknown u can be easily approximated through a simple post-processing procedure based on the equilibrium equation. We prove unique solvability of the resulting continuous and discrete problems by means of the generalized Lax-Milgram lemma. In particular, the well-posedness, stability and convergence of the Galerkin scheme can be achieved through a sufficiently small mesh-size assumption. Next, we derive a reliable and efficient residual-based a posteriori error estimator for the conforming method. The proof of reliability makes use of the global inf-sup condition, Helmholtz decomposition, and the local approximation properties of the Clement interpolant and Raviart-Thomas operator. On the other hand, inverse inequalities, the localization technique based on element-bubble and edge-bubble functions, and known results from previous works, are the main tools for proving the efficiency of the estimator. Finally, some numerical results confirming the good performance of the method and the theoretical properties of the a posteriori error estimator, and illustrating the capability of the corresponding adaptive algorithm to localize the singularities of the solution, are reported.