An expanded mixed finite element approach via a dual-dual formulation
and the minimum residual method
Gabriel N. Gatica, Norbert Heuer
J. Comput. Appl. Math. 132 (2), 371-385, 2001.
We apply an expanded mixed finite element method, which introduces the gradient
as a third explicit unknown, to solve a linear second order elliptic equation
in divergence form. Instead of using the standard dual form, we show that the
corresponding variational formulation can be written as a dual-dual
operator equation. We establish existence and uniqueness of solution for the
continuous and discrete formulations, and provide the corresponding error
analysis by using Raviart-Thomas elements. A generalization of the classical
Babuska-Brezzi theory to this kind of formulations is the main tool of our
analysis. In addition, we show that the corresponding dual-dual linear system
can be efficiently solved by a preconditioned minimum residual method. Some
numerical results, illustrating this fact and the rate of convergence of the
mixed finite element method, are also provided.