Abstract
Nitsche’s method is a penalty-based method to weakly enforce boundary conditions in the finite element method. In this paper, we present a penalty free version of Nitsche’s method to impose interface coupling in the framework of unfitted domain decomposition. Unfitted domain decomposition is understood in the sense that the interface between the domains can cross elements of the mesh arbitrarily. The pure diffusion problem with discontinuous material parameters is considered for the theoretical study, we show the convergence of the L 2 and H 1-error for high contrast in the diffusivities. Then, we give the corresponding numerical results for the pure diffusion problem, additionally we consider the Stokes problem. We compare the performance of the penalty free method with the more classical symmetric and nonsymmetric Nitsche’s methods for different cases, including for the error generated in the interface fluxes.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Arnold, D.N.: An interior penalty finite element method with discontinuous elements. SIAM J. Numer. Anal. 19(4), 742–760 (1982). http://dx.doi.org/10.1137/0719052
Babuška, I.: The finite element method with Lagrangian multipliers. Numer. Math. 20, 179–192 (1972/73)
Becker, R., Hansbo, P., Stenberg, R.: A finite element method for domain decomposition with non-matching grids. M2AN Math. Model. Numer. Anal. 37(2), 209–225 (2003). http://dx.doi.org/10.1051/m2an:2003023
Boiveau, T.: Penalty-free Nitsche method for interface problems in computational mechanics. Ph.D. thesis, University College London (2016)
Boiveau, T., Burman, E.: A penalty-free Nitsche method for the weak imposition of boundary conditions in compressible and incompressible elasticity. IMA J. Numer. Anal. 36(2), 770–795 (2016). http://dx.doi.org/10.1093/imanum/drv042
Boiveau, T., Burman, E., Claus, S., Larson, M.G.: Fictitious domain method with boundary value correction using penalty-free Nitsche method (2016). https://arxiv.org/abs/1610.04482
Brezzi, F.: On the existence, uniqueness and approximation of saddle-point problems arising from lagrangian multipliers. ESAIM: Math. Model. Numer. Anal. (Modélisation Mathématique et Analyse Numérique) 8(R2), 129–151 (1974). http://eudml.org/doc/193255
Burman, E.: A penalty free nonsymmetric Nitsche-type method for the weak imposition of boundary conditions. SIAM J. Numer. Anal. 50(4), 1959–1981 (2012). http://dx.doi.org/10.1137/10081784X
Burman, E.: Ghost penalty. C. R. Math. Acad. Sci. Paris 348(21–22), 1217–1220 (2010). http://dx.doi.org/10.1016/j.crma.2010.10.006
Chu, C.C., Graham, I.G., Hou, T.Y.: A new multiscale finite element method for high-contrast elliptic interface problems. Math. Comput. 79(272), 1915–1955 (2010). http://dx.doi.org/10.1090/S0025-5718-2010-02372-5
Freund, J., Stenberg, R.: On weakly imposed boundary conditions for second order problems. In: Proceedings of the Ninth International Conference on Finite Elements in Fluids, pp. 327–336. Università di Padova, Padova (1995)
Hansbo, A., Hansbo, P.: An unfitted finite element method, based on Nitsche’s method, for elliptic interface problems. Comput. Methods Appl. Mech. Eng. 191(47–48), 5537–5552 (2002). http://dx.doi.org/10.1016/S0045-7825(02)00524-8
Hansbo, A., Hansbo, P., Larson, M.G.: A finite element method on composite grids based on Nitsche’s method. M2AN Math. Model. Numer. Anal. 37(3), 495–514 (2003). http://dx.doi.org/10.1051/m2an:2003039
Hansbo, P.: Nitsche’s method for interface problems in computational mechanics. GAMM-Mitt. 28(2), 183–206 (2005). http://dx.doi.org/10.1002/gamm.201490018
Hansbo, P., Larson, M.G., Zahedi, S.: A cut finite element method for a Stokes interface problem. Appl. Numer. Math. 85(0), 90–114 (2014). http://dx.doi.org/10.1016/j.apnum.2014.06.009. http://www.sciencedirect.com/science/article/pii/S0168927414001184
Hansbo, P., Lovadina, C., Perugia, I., Sangalli, G.: A Lagrange multiplier method for elliptic interface problems using non-matching meshes. In: Applied and Industrial Mathematics in Italy, Series on Advances in Mathematics for Applied Sciences, vol. 69, pp. 360–370. World Scientific, Hackensack (2005). http://dx.doi.org/10.1142/9789812701817_0033
Hughes, T.J.R., Engel, G., Mazzei, L., Larson, M.G.: A comparison of discontinuous and continuous Galerkin methods based on error estimates, conservation, robustness and efficiency. In: Discontinuous Galerkin Methods (Newport, RI, 1999), Lecture Notes in Computational Science and Engineering, vol. 11, pp. 135–146. Springer, Berlin (2000). http://dx.doi.org/10.1007/978-3-642-59721-3_9
Kirchhart, M., Gross, S., Reusken, A.: Analysis of an xfem discretization for Stokes interface problems. Preprint 420
Larson, M.G., Massing, A.: L2-error estimates for finite element approximations of boundary fluxes. arXiv preprint arXiv:1401.6994 (2014)
Massing, A., Larson, M.G., Logg, A., Rognes, M.E.: A stabilized Nitsche fictitious domain method for the stokes problem. J. Sci. Comput. 61(3), 604–628 (2014). http://dx.doi.org/10.1007/s10915-014-9838-9
Nitsche, J.: Über ein variationsprinzip zur lösung von Dirichlet-problemen bei verwendung von teilräumen, die keinen randbedingungen unterworfen sind. Abh. Math. Semin. Univ. Hambg. 36(1), 9–15 (1971). http://dx.doi.org/10.1007/BF02995904
Pitkäranta, J.: Boundary subspaces for the finite element method with Lagrange multipliers. Numer. Math. 33(3), 273–289 (1979). http://dx.doi.org/10.1007/BF01398644
Pitkäranta, J.: Local stability conditions for the Babuška method of Lagrange multipliers. Math. Comput. 35(152), 1113–1129 (1980). http://dx.doi.org/10.2307/2006378
Stenberg, R.: On some techniques for approximating boundary conditions in the finite element method. J. Comput. Appl. Math. 63(1–3), 139–148 (1995). http://dx.doi.org/10.1016/0377-0427(95)00057-7. International Symposium on Mathematical Modelling and Computational Methods Modelling 94 (Prague, 1994)
Acknowledgements
This work received funding from EPSRC (award number EP/J002313/2) which is gratefully acknowledged. The Author, S. Claus, gratefully acknowledges the financial support provided by the Welsh Government and Higher Education Funding Council for Wales through the Sêr Cymru National Research Network in Advanced Engineering and Materials.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2017 Springer International Publishing AG
About this paper
Cite this paper
Boiveau, T., Burman, E., Claus, S. (2017). Penalty-Free Nitsche Method for Interface Problems. In: Bordas, S., Burman, E., Larson, M., Olshanskii, M. (eds) Geometrically Unfitted Finite Element Methods and Applications. Lecture Notes in Computational Science and Engineering, vol 121. Springer, Cham. https://doi.org/10.1007/978-3-319-71431-8_6
Download citation
DOI: https://doi.org/10.1007/978-3-319-71431-8_6
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-71430-1
Online ISBN: 978-3-319-71431-8
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)