Hostname: page-component-cd9895bd7-8ctnn Total loading time: 0 Render date: 2024-12-23T18:51:48.604Z Has data issue: false hasContentIssue false
  • English
  • Français

A NEW MINIMIZATION PRINCIPLE FOR THE POISSON EQUATION LEADING TO A FLEXIBLE FINITE ELEMENT APPROACH

Part of: Approximations and expansions Numerical analysis: Ordinary differential equations Numerical approximation and computational geometry

Published online by Cambridge University Press:  03 October 2017

B. P. LAMICHHANE Open the ORCID record for B. P. LAMICHHANE [Opens in a new window]
B. P. LAMICHHANE*
Affiliation:
School of Mathematical and Physical Sciences, University of Newcastle, Callaghan, NSW 2308, Australia email [email protected]
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

A new minimization principle for the Poisson equation using two variables – the solution and the gradient of the solution – is introduced. This principle allows us to use any conforming finite element spaces for both variables, where the finite element spaces do not need to satisfy the so-called inf–sup condition. A numerical example demonstrates the superiority of this approach.

Keywords

Poisson equationminimization principlemixed finite element methoda priori error estimate

MSC classification

Primary: 65D15: Algorithms for functional approximation
Secondary: 65L60: Finite elements, Rayleigh-Ritz, Galerkin and collocation methods 41A15: Spline approximation
Type
Research Article
Information
The ANZIAM Journal , Volume 59 , Issue 2 , October 2017 , pp. 232 - 239
Copyright
© 2017 Australian Mathematical Society 

References

Achdou, Y., Bernardi, C. and Coquel, F., “A priori and a posteriori analysis of finite volume discretizations of Darcy’s equations”, Numer. Math. 96 (2003) 17–42; doi:10.1007/s00211-002-0436-7.Google Scholar
Bochev, P. B. and Dohrmann, C. R., “A computational study of stabilized, low-order $C^{0}$ finite element approximations of Darcy equations”, Comput. Mech. 38 (2006) 323–333; doi:10.1007/s00466-006-0036-y.Google Scholar
Bochev, P. B. and Gunzburger, M., “Finite element methods of least-squares type”, SIAM Rev. 40 (1998) 789–837; doi:10.1137/S0036144597321156.Google Scholar
Bochev, P. B. and Gunzburger, M., “On least-squares finite element methods for the Poisson equation and their connection to the Dirichlet and Kelvin principles”, SIAM J. Numer. Anal. 43 (2005) 340–362; doi:10.1137/S003614290443353X.CrossRefGoogle Scholar
Bochev, P. B. and Gunzburger, M., Least-squares finite element methods (Springer, New York, 2009).Google Scholar
Braess, D., Finite elements. Theory, fast solvers, and applications in solid mechanics, 2nd edn (Cambridge University Press, Cambridge, 2001).Google Scholar
Brenner, S. C. and Scott, L. R., The mathematical theory of finite element methods (Springer, New York, 1994).Google Scholar
Brezzi, F., Hughes, T. J. R., Marini, L. D. and Masud, A., “Mixed discontinuous Galerkin methods for Darcy flow”, J. Sci. Comput. 22–23 (2005) 119–145; doi:10.1007/s10915-004-4150-8.Google Scholar
Ciarlet, P. G., The finite element method for elliptic problems (North-Holland, Amsterdam, 1978).Google Scholar
Jiang, B., The least-squares finite element method: theory and applications in computational fluid dynamics and electromagnetics (Springer, Berlin, 1998).Google Scholar
Lamichhane, B. P., “Mixed finite element methods for the Poisson equation using biorthogonal and quasi-biorthogonal systems”, Adv. Numer. Anal. 2013 (2013); Article 189045 1–9 (electronic); doi:10.1155/2013/189045.Google Scholar
Lamichhane, B. P., Reddy, B. D. and Wohlmuth, B. I., “Convergence in the incompressible limit of finite element approximations based on the Hu–Washizu formulation”, Numer. Math. 104 (2006) 151–175; doi:10.1007/s00211-006-0014-5.Google Scholar
Mardal, K. A., Tai, X.-C. and Winther, R., “A robust finite element method for Darcy–Stokes flow”, SIAM J. Numer. Anal. 40 (2002) 1605–1631; doi:10.1137/S0036142901383910.Google Scholar
Stenberg, R., “A nonstandard mixed finite element family”, Numer. Math. 115 (2010) 131–139; doi:10.1007/s00211-009-0272-0.Google Scholar
Urquiza, J. M., N’Dri, D., Garon, A. and Delfour, M. C., “A numerical study of primal mixed finite element approximations of Darcy equations”, Comm. Numer. Methods Engrg. 22 (2006) 901–915; doi:10.1002/cnm.859.Google Scholar