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 [1–8]. However, all of them are based on the two-field formulation of the
References
[1]
Y. Achdou, C. Bernardi, and F. Coquel, “A priori and a posteriori analysis of finite volume discretizations of Darcy's equations,” Numerische Mathematik, vol. 96, no. 1, pp. 17–42, 2003.
[2]
P. B. Bochev and C. R. Dohrmann, “A computational study of stabilized, low-order finite element approximations of Darcy equations,” Computational Mechanics, vol. 38, no. 4-5, pp. 310–333, 2006.
[3]
D. Braess, Finite Elements. Theory, Fast Solver, and Applications in Solid Mechanics, Cambridge University Press, Cambridge, UK, 2nd edition, 2001.
[4]
F. Brezzi and M. Fortin, Mixed and Hybrid Finite Element Methods, Springer, New York, NY, USA, 1991.
[5]
F. Brezzi, T. J. R. Hughes, L. D. Marini, and A. Masud, “Mixed discontinuous Galerkin methods for Darcy flow,” Journal of Scientific Computing, vol. 22-23, pp. 119–145, 2005.
[6]
K. A. Mardal, X.-C. Tai, and R. Winther, “A robust finite element method for Darcy-Stokes flow,” SIAM Journal on Numerical Analysis, vol. 40, no. 5, pp. 1605–1631, 2002.
[7]
A. Masud and T. J. R. Hughes, “A stabilized mixed finite element method for Darcy flow,” Computer Methods in Applied Mechanics and Engineering, vol. 191, no. 39-40, pp. 4341–4370, 2002.
[8]
J. M. Urquiza, D. N'Dri, A. Garon, and M. C. Delfour, “A numerical study of primal mixed finite element approximations of Darcy equations,” Communications in Numerical Methods in Engineering with Biomedical Applications, vol. 22, no. 8, pp. 901–915, 2006.
[9]
B. P. Lamichhane and E. P. Stephan, “A symmetric mixed finite element method for nearly incompressible elasticity based on biorthogonal systems,” Numerical Methods for Partial Differential Equations, vol. 28, no. 4, pp. 1336–1353, 2012.
[10]
S. C. Brenner and L. R. Scott, The Mathematical Theory of Finite Element Methods, Springer, New York, NY, USA, 1994.
[11]
B. P. Lamichhane, B. D. Reddy, and B. I. Wohlmuth, “Convergence in the incompressible limit of finite element approximations based on the Hu-Washizu formulation,” Numerische Mathematik, vol. 104, no. 2, pp. 151–175, 2006.
[12]
J. C. Simo and M. S. Rifai, “A class of mixed assumed strain methods and the method of incompatible modes,” International Journal for Numerical Methods in Engineering, vol. 29, no. 8, pp. 1595–1638, 1990.
[13]
K. Washizu, Variational Methods in Elasticity and Plasticity, Pergamon Press, 3rd edition, 1982.
[14]
D. N. Arnold and F. Brezzi, “Some new elements for the Reissner-Mindlin plate model,” in Boundary Value Problems for Partial Differerntial Equations and Applications, pp. 287–292, Masson, Paris, France, 1993.
[15]
P. G. Ciarlet, The Finite Element Method for Elliptic Problems, North-Holland, Amsterdam, The Netherlands, 1978.
[16]
B. P. Lamichhane, Higher order mortar finite elements with dual lagrange multiplier spaces and applications [Ph.D. thesis], University of Stuttgart, 2006.
[17]
B. P. Lamichhane, “A stabilized mixed finite element method for the biharmonic equation based on biorthogonal systems,” Journal of Computational and Applied Mathematics, vol. 235, no. 17, pp. 5188–5197, 2011.
[18]
B. P. Lamichhane, “A mixed finite element method for the biharmonic problem using biorthogonal or quasi-biorthogonal systems,” Journal of Scientific Computing, vol. 46, no. 3, pp. 379–396, 2011.
[19]
B. P. Lamichhane, S. Roberts, and L. Stals, “A mixed finite element discretisation of thin-plate splines,” ANZIAM Journal, vol. 52, pp. C518–C534, 2011, Proceedings of the 15th Biennial Computational Techniques and Applications Conference.
[20]
B. P. Lamichhane and B. I. Wohlmuth, “Biorthogonal bases with local support and approximation properties,” Mathematics of Computation, vol. 76, no. 257, pp. 233–249, 2007.
[21]
A. Galántai, Projectors and Projection Methods, vol. 6, Kluwer Academic Publishers, Dordrecht, The Netherlands, 2004.
[22]
D. B. Szyld, “The many proofs of an identity on the norm of oblique projections,” Numerical Algorithms, vol. 42, no. 3-4, pp. 309–323, 2006.
[23]
C. Kim, R. D. Lazarov, J. E. Pasciak, and P. S. Vassilevski, “Multiplier spaces for the mortar finite element method in three dimensions,” SIAM Journal on Numerical Analysis, vol. 39, no. 2, pp. 519–538, 2001.
[24]
D. N. Arnold, F. Brezzi, and M. Fortin, “A stable finite element for the Stokes equations,” Calcolo, vol. 21, no. 4, pp. 337–344, 1984.
[25]
B. P. Lamichhane, “A mixed finite element method for non-linear and nearly incompressible elasticity based on biorthogonal systems,” International Journal for Numerical Methods in Engineering, vol. 79, no. 7, pp. 870–886, 2009.
[26]
B. I. Wohlmuth, Discretization Methods and Iterative Solvers Based on Domain Decomposition, vol. 17 of Lecture Notes in Computational Science and Engineering, Springer, Heidelberg, Germany, 2001.
[27]
M. Ainsworth and J. T. Oden, A Posteriori Error Estimation in Finite Element Analysis, Wiley-Interscience, New York, NY, USA, 2000.
