A three-field formulation of the Poisson problem with Nitsche approach

Ilyas, Muhammad ; Lamichhane, Bishnu P.

ANZIAM Journal, Tome 59 (2018), / Harvested from Australian Mathematical Society

Résumé

We modify a three-field formulation of the Poisson problem with Nitsche approach for approximating Dirichlet boundary conditions. Nitsche approach allows us to weakly impose Dirichlet boundary condition but still preserves the optimal convergence. We use a biorthogonal system for efficient numerical computation and introduce a stabilisation term so that the problem is coercive on the whole space. Numerical examples are presented to verify the algebraic formulation of the problem. References J. P. Aubin. Approximation of elliptic boundary-value problems, volume XXVI of Pure and Applied Mathematics. Wiley-Interscience, New York, 1972. doi:10.1137/1016069. I. Babuska. The finite element method with penalty. Mathematics of Computation, 27(122):221–228, 1973. doi:10.2307/2005611. R. Becker, E. Burman, and P. Hansbo. A Nitsche extended finite element method for incompressible elasticity with discontinuous modulus of elasticity. Computer Methods in Applied Mechanics and Engineering, 198(41):3352 – 3360, 2009. doi:10.1016/j.cma.2009.06.017. D. Braess. Finite Elements: Theory, Fast Solvers, and Applications in Solid Mechanics. Cambridge University Press, Cambridge, UK, 3rd edition edition, 2007. doi:10.1017/CBO9780511618635. P. G. Ciarlet. The finite element method for elliptic problems. North Holland, Amsterdam, 1978. doi:10.1137/1.9780898719208. R. Codina and J. Baiges. Weak imposition of essential boundary conditions in the finite element approximation of elliptic problems with non-matching meshes. International Journal for Numerical Methods in Engineering, 104(7):624–654, 2015. doi:10.1002/nme.4815. P. Hansbo. Nitsche's method for interface problems in computational mechanics. GAMM-Mitteilungen, 28(2):183–206, 2005. doi:10.1002/gamm.201490018. M. Ilyas and B. P. Lamichhane. A stabilised mixed finite element method for the Poisson problem based on a three-field formulation. In Proceedings of the 12th Biennial Engineering Mathematics and Applications Conference, EMAC-2015, volume 57 of ANZIAM J., pages C177–C192, September 2016. doi:10.21914/anziamj.v57i0.10356 . M. Ilyas and B. P. Lamichhane. A three-field formulation of the poisson problem with Nitsche approach. arXiv:1711.05961, November 2017. A. Johansson, M. Garzon, and J. A. Sethian. A three-dimensional coupled Nitsche and level set method for electrohydrodynamic potential flows in moving domains. Journal of Computational Physics, 309:88 – 111, 2016. doi:10.1016/j.jcp.2015.12.026. M. Juntunen and R. Stenberg. On a mixed discontinuous galerkin method. Electronic Transactions on Numerical Analysis, 32:17–32, 2008. M. Juntunen and R. Stenberg. Nitsche's method for general boundary conditions. Mathematics of Computation, 78:1353–1374, 2009. doi:10.1090/S0025-5718-08-02183-2. B. P. Lamichhane, A. T. McBride, and B. D. Reddy. A finite element method for a three-field formulation of linear elasticity based on biorthogonal systems. Computer Methods in Applied Mechanics and Engineering, 258:109–117, 2013. doi:10.1016/j.cma.2013.02.008. B.P. Lamichhane and B.I. Wohlmuth. A quasi-dual Lagrange multiplier space for Serendipity Mortar finite elements in 3D. \(M^2AN\), 38:73–92, 2004. doi:10.1051/m2an:2004004. J. Nitsche. Uber ein Variationsprinzip zur Losung von Dirichlet-Problemen bei Verwendung von Teilraumen, die keinen Randbedingungen unterworfen sind. Abhandlungen aus dem Mathematischen Seminar der Universitat Hamburg, 36(1):9–15, Jul 1971. T. J. Truster. A stabilized, symmetric Nitsche method for spatially localized plasticity. Computational Mechanics, 57(1):75–103, Jan 2016. doi:10.1007/s00466-015-1222-6.

Détail

Publié le : 2018-01-01
DOI : https://doi.org/10.21914/anziamj.v59i0.12645