Abstract
There are many physical problems such as multiphase flows and fluid-structure interactions whose solutions are piecewise smooth but may have discontinuity across some curved interfaces. The direct application of standard finite element method may not perform well. In this paper, we study some special finite element methods for this type of problems. For simplicity of exposition, we consider the case that there is only one interface which is smooth. Let \(\varOmega,\varOmega _{1} \subset \mathbb{R}^{2}\) be two bounded domains with \(\varOmega _{1} \subset \varOmega\). We assume that Γ = ∂ Ω 1 is sufficiently smooth, and \(\varGamma \cap \partial \varOmega =\emptyset\). To be focused on the influence of Γ, we assume \(\varOmega = (-1,1)^{2}\).
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
Notes
- 1.
Both authors are supported by the Department of Energy (DOE) Grants DE-SC0006903 and DE-SC0009249 (through the Applied Mathematics Program within the DOE Office of Advanced Scientific Computing Research (ASCR) as part of the Collaboratory on Mathematics for Mesoscopic Modeling of Materials (CM4)) and the Center for Computational Mathematics and Applications of the Pennsylvania State University. The second author is also supported by the NSFC Grant 11101415 and SRF for ROCS by SEM of P. R. China.
References
G. Acosta, R.G. Durán, The maximum angle condition for mixed and nonconforming elements: application to the Stokes equations. SIAM J. Numer. Anal. 37(1), 18–36 (1999) (electronic). ISSN 0036-1429. doi:10.1137/S0036142997331293. http://dx.doi.org/10.1137/S0036142997331293
J.M. Arrieta, A. Rodríguez-Bernal, J.D. Rossi, The best Sobolev trace constant as limit of the usual Sobolev constant for small strips near the boundary. Proc. R. Soc. Edinb. Sect. A 138(2), 223–237 (2008). ISSN 0308-2105. doi:10.1017/S0308210506000813. http://dx.doi.org/10.1017/S0308210506000813
J.H. Bramble, J.T. King, A finite element method for interface problems in domains with smooth boundaries and interfaces. Adv. Comput. Math. 6(2), 109–138 (1996). ISSN 1019-7168. doi:10.1007/BF02127700. http://dx.doi.org/10.1007/BF02127700
F. Brezzi, M. Fortin, Mixed and Hybrid Finite Element Methods. Springer Series in Computational Mathematics, vol. 15 (Springer, New York, 1991). ISBN 0-387-97582-9. doi:10.1007/978-1-4612-3172-1. http://dx.doi.org/10.1007/978-1-4612-3172-1
L. Chen, R.H. Nochetto, J. Xu, Optimal multilevel methods for graded bisection grids. Numer. Math. 120(1), 1–34 (2012). ISSN 0029-599X. doi:10.1007/s00211-011-0401-4. http://dx.doi.org/10.1007/s00211-011-0401-4
Z. Chen, J. Zou, Finite element methods and their convergence for elliptic and parabolic interface problems. Numer. Math. 79(2), 175–202 (1998). ISSN 0029-599X. doi:10.1007/s002110050336. http://dx.doi.org/10.1007/s002110050336.
Z. Chen, Z. Wu, Y. Xiao, An adaptive immersed finite element method with arbitrary lagrangian-eulerian scheme for parabolic equations in variable domains. Int J. Numer. Anal. Model. 12(3), 567–591 (2015)
J. Li, J.M. Melenk, B. Wohlmuth, J. Zou, Optimal a priori estimates for higher order finite elements for elliptic interface problems. Appl. Numer. Math. 60(1–2), 19–37 (2010). ISSN 0168-9274. doi:10.1016/j.apnum.2009.08.005. http://dx.doi.org/10.1016/j.apnum.2009.08.005
L.R. Scott, S. Zhang, Finite element interpolation of nonsmooth functions satisfying boundary conditions. Math. Comput. 54(190), 483–493 (1990). ISSN 0025-5718. doi:10.2307/2008497. http://dx.doi.org/10.2307/2008497
J. Xu, Estimate of the convergence rate of finite element solutions to elliptic equations of second order with discontinuous coefficients (in chinese). Natural Science Journal of Xiangtan University (1982), pp. 1–5
J. Xu, Iterative methods by space decomposition and subspace correction. SIAM Rev. 34(4), 581–613 (1992) ISSN 0036-1445. doi:10.1137/1034116. http://dx.doi.org/10.1137/1034116
J. Xu, Estimate of the convergence rate of finite element solutions to elliptic equations of second order with discontinuous coefficients. arXiv preprint arXiv:1311.4178 (2013)
J. Xu, Y. Zhu, Uniform convergent multigrid methods for elliptic problems with strongly discontinuous coefficients. Math. Models Methods Appl. Sci. 18(1), 77–105 (2008) ISSN 0218-2025. doi:10.1142/S0218202508002619. http://dx.doi.org/10.1142/S0218202508002619
J. Xu, L. Zikatanov, The method of alternating projections and the method of subspace corrections in Hilbert space. J. Am. Math. Soc. 15(3), 573–597 (2002). ISSN 0894-0347. doi:10.1090/S0894-0347-02-00398-3. http://dx.doi.org/10.1090/S0894-0347-02-00398-3
J. Xu, L. Chen, R.H. Nochetto, Optimal multilevel methods for H(grad), H(curl), and H(div) systems on graded and unstructured grids, in Multiscale, Nonlinear and Adaptive Approximation (Springer, Berlin, 2009), pp. 599–659. doi:10.1007/978-3-642-03413-8_14. http://dx.doi.org/10.1007/978-3-642-03413-8_14
Acknowledgement
The authors would like to thank Dr. Xiaozhe Hu for his help on the numerical examples.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2016 Springer International Publishing Switzerland
About this paper
Cite this paper
Xu, J., Zhang, S. (2016). Optimal Finite Element Methods for Interface Problems. In: Dickopf, T., Gander, M., Halpern, L., Krause, R., Pavarino, L. (eds) Domain Decomposition Methods in Science and Engineering XXII. Lecture Notes in Computational Science and Engineering, vol 104. Springer, Cham. https://doi.org/10.1007/978-3-319-18827-0_7
Download citation
DOI: https://doi.org/10.1007/978-3-319-18827-0_7
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-18826-3
Online ISBN: 978-3-319-18827-0
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)