%0 Journal Article
%T Mixed Finite Element Methods for the Poisson Equation Using Biorthogonal and Quasi-Biorthogonal Systems
%A Bishnu P. Lamichhane
%J Advances in Numerical Analysis
%D 2013
%I Hindawi Publishing Corporation
%R 10.1155/2013/189045
%X We introduce two three-field mixed formulations for the Poisson equation and propose finite 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 efficient 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 finite element approximations. 1. Introduction In many practical situations, it is important to compute dual variables of partial differential 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 finite element approach these variables should be obtained a posteriori by differentiation, which will result in a loss of accuracy. In these situations, a mixed method is often preferred as these variables can be directly computed using a mixed method. In this paper, we introduce two mixed finite 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 equation as a new unknown and writing an additional variational equation in terms of a Lagrange multiplier. This gives rise to two additional vector unknowns: the gradient of the solution and the Lagrange multiplier. In order to obtain an efficient 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 system 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 definite formulation. The positive definite 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. The first formulation is discretized by using a quasi-biorthogonal system, whereas the second one, which is a stabilized version of the first one, is discretized using a biorthogonal system. There are many mixed finite element methods for the Poisson equation [1¨C8]. However, all of them are based on the two-field formulation of the
%U http://www.hindawi.com/journals/ana/2013/189045/