Hindawi Publishing Corporation Advances in Numerical Analysis Volume 2013, Article ID 189045, 9 pages http://dx.doi.org/10.1155/2013/189045 Research Article Mixed Finite Element Methods for the Poisson Equation Using Biorthogonal and Quasi-Biorthogonal Systems Bishnu P. Lamichhane School of Mathematical and Physical Sciences, University of Newcastle, Callaghan, NSW 2308, Australia Correspondence should be addressed to Bishnu P. Lamichhane; blamichha@gmail.com Received 10 October 2012; Revised 20 February 2013; Accepted 25 February 2013 Academic Editor: Norbert Heuer Copyright © 2013 Bishnu P. Lamichhane. Tis is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. We introduce two three-feld mixed formulations for the Poisson equation and propose fnite element methods for their approximation. Both mixed formulations are obtained by introducing a weak equation for the gradient of the solution by means of a Lagrange multiplier space. Two efcient numerical schemes are proposed based on using a pair of bases for the gradient of the solution and the Lagrange multiplier space forming biorthogonal and quasi-biorthogonal systems, respectively. We also establish an optimal a priori error estimate for both fnite element approximations. 1. Introduction In many practical situations, it is important to compute dual variables of partial diferential equations more accurately. For example, the gradient of the solution is the dual variable in case of the Poisson equation, whereas the stress or pressure variable is the dual variable in case of elasticity equation. Working with the standard fnite element approach these variables should be obtained a posteriori by diferentiation, which will result in a loss of accuracy. In these situations, a mixed method is ofen preferred as these variables can be directly computed using a mixed method. In this paper, we introduce two mixed fnite element methods for the Poisson equation using biorthogonal or quasi-biorthogonal systems. Both formulations are obtained by introducing the gradient of the solution of Poisson equa- tion as a new unknown and writing an additional variational equation in terms of a Lagrange multiplier. Tis gives rise to two additional vector unknowns: the gradient of the solution and the Lagrange multiplier. In order to obtain an efcient numerical scheme, we carefully choose a pair of bases for the space of the gradient of the solution and the Lagrange multiplier space in the discrete setting. Choosing the pair of bases forming a biorthogonal or quasi-biorthogonal sys- tem for these two spaces, we can eliminate the degrees of freedom associated with the gradient of the solution and the Lagrange multiplier and arrive at a positive defnite formulation. Te positive defnite formulation involves only the degrees of freedom associated with the solution of the Poisson equation. Hence a reduced system is obtained, which is easy to solve. Te frst formulation is discretized by using a quasi-biorthogonal system, whereas the second one, which is a stabilized version of the frst one, is discretized using a biorthogonal system. Tere are many mixed fnite element methods for the Poisson equation [1–8]. However, all of them are based on the two-feld formulation of the Poisson equation and hence are not amenable to the application of the biorthogonal and quasi-biorthogonal systems. We need a three-feld for- mulation to apply the biorthogonal and quasi-biorthogonal systems which leads to a symmetric formulation (see [9] for a three-feld formulation in linear elasticity). Te use of biorthogonal and quasi-biorthogonal systems allows an easy static condensation of the auxiliary variables (gradi- ent of the solution and Lagrange multiplier) leading to a reduced linear system. Tese variables can be recovered just by inverting a diagonal matrix. Terefore, in this paper we present three-feld formulations of the Poisson equation to apply the biorthogonal and quasi-biorthogonal systems. Te structure of the rest of the paper is as follows. In Section 2, we introduce our three-feld formulations and