Methods and Algorithms for Scientific Computing

A Multigrid Method for Unfitted Finite Element Discretizations of Elliptic Interface Problems

Abstract

We consider discrete Poisson interface problems resulting from linear unfitted finite elements, also called cut finite elements. Three of these unfitted finite element methods, known from the literature, are studied. Two of these are suitable only for small jumps in the diffusion coefficient, and the third one has a robustness property that makes it appropriate also for interface problems with large coefficient jumps. All three methods rely on Nitsche's method to incorporate the interface conditions. The main topic of the paper is the development of a multigrid method, based on a novel prolongation operator for the unfitted finite element space and an interface smoother that is designed to yield robustness for large jumps in the diffusion coefficients. Numerical results are presented which illustrate efficiency of this multigrid method and demonstrate its robustness properties with respect to variation of the mesh size, location of the interface, and contrast in the diffusion coefficients.

Keywords

  1. geometric multigrid
  2. unfitted finite elements
  3. interface problem
  4. Nitsche
  5. CutFEM

MSC codes

  1. 65N30
  2. 65F10
  3. 65N22
  4. 65N55

Get full access to this article

View all available purchase options and get full access to this article.

References

1.
L. Adams and T. P. Chartier, New geometric immersed interface multigrid solvers, SIAM J. Sci. Comput., 25 (2004), pp. 1516--1533.
2.
L. Adams and Z. Li, The immersed interface/multigrid methods for interface problems, SIAM J. Sci. Comput., 24 (2002), pp. 463--479.
3.
C. Annavarapu, M. Hautefeuille, and J. Dolbow, A robust Nitsche's formulation for interface problems, Comput. Methods Appl. Mech. Engrg., 225 (2012), pp. 44--54.
4.
B. Ayuso de Dios, M. Holst, Y. Zhu, and L. Zikatanov, Multilevel preconditioners for discontinuous Galerkin approximations of elliptic problems with jump coefficients, Math. Comp., 83 (2014), pp. 1083--1120.
5.
S. Badia and F. Verdugo, Robust and scalable domain decomposition solvers for unfitted finite element methods, J. Comput. Appl. Math., 344 (2018), pp. 740--759.
6.
N. Barrau, R. Becker, E. Dubach, and R. Luce, A robust variant of NXFEM for the interface problem, C. R. Math. Acad. Sci. Paris, 350 (2012), pp. 789--792.
7.
L. Berger-Vergiat, H. Waisman, B. Hiriyur, R. Tuminaro, and D. Keyes, Inexact Schwarz-algebraic multigrid preconditioners for crack problems modeled by extended finite element methods, Internat. J. Numer. Methods Engrg., 90 (2012), pp. 311--328.
8.
J. H. Bramble, Multigrid Methods, CRC Press, Boca Raton, FL, 1993.
9.
E. Burman, La pénalisation fantôme, C. R. Math. Acad. Sci. Paris, 348 (2010), pp. 1217--1220.
10.
E. Burman, S. Claus, P. Hansbo, M. Larson, and A. Massing, CutFEM: Discretizing geometry and partial differential equations, Internat. J. Numer. Methods Engrg., 104 (2015), pp. 472--501.
11.
E. Burman and P. Zunino, Numerical approximation of large contrast problems with the unfitted Nitsche method, in Frontiers in Numerical Analysis - Durham 2010, J. Blowey and M. Jensen, eds., Springer-Verlag, Berlin, 2011, pp. 227--282.
12.
T. Chen and J. Strain, Piecewise-polynomial discretization and Krylov-accelerated multigrid for elliptic interface problems, J. Comput. Phys., 227 (2008), pp. 7503--7542.
13.
F. de Prenter, C. Verhoosel, and H. van Brummelen, Preconditioning immersed isogeometric finite element methods with application to flow problems, Comput. Methods Appl. Mech. Engrg., 348 (2019), pp. 604--631.
14.
F. de Prenter, C. V. Verhoosel, G. van Zwieten, and E. H. van Brummelen, Condition number analysis and preconditioning of the finite cell method, Comput. Methods Appl. Mech. Engrg., 316 (2017), pp. 297--327.
15.
T.-P. Fries and T. Belytschko, The extended/generalized finite element method: An overview of the method and its applications, Internat. J. Numer. Methods Engrg., 84 (2010), pp. 253--304.
16.
A. Gerstenberger and R. Tuminaro, An algebraic multigrid approach to solve extended finite element method based fracture problems, Internat. J. Numer. Methods Engrg., 94 (2013), pp. 248--272.
17.
M. Griebel and P. Oswald, On the abstract theory of additive and multiplicative Schwarz algorithms, Numer. Math., 70 (1995), pp. 163--180.
18.
A. Hansbo and P. Hansbo, An unfitted finite element method, based on Nitsche's method, for elliptic interface problems, Comput. Methods Appl. Mech. Engrg., 191 (2002), pp. 5537--5552.
19.
B. Janssen and G. Kanschat, Adaptive multilevel methods with local smoothing for H^1- and H^curl-conforming high order finite element methods, SIAM J. Sci. Comput., 33 (2011), pp. 2095--2114.
20.
G. Jo and D. Y. Kwak, Geometric multigrid algorithms for elliptic interface problems using structured grids, Numer. Algorithms, (2018), pp. 1--25.
21.
C. Lehrenfeld, On a Space-Time Extended Finite Element Method for the Solution of a Class of Two-Phase Mass Transport Problems, Ph.D. thesis, Universitätsbibliothek der RWTH Aachen, Aachen, Germany, 2015.
22.
C. Lehrenfeld, High order unfitted finite element methods on level set domains using isoparametric mappings, Comput. Methods Appl. Mech. Engrg., 300 (2016), pp. 716--733.
23.
C. Lehrenfeld, Removing the Stabilization Parameter in Fitted and Unfitted Symmetric Nitsche Formulations, arXiv preprint arXiv:1603.00617, 2016.
24.
C. Lehrenfeld and A. Reusken, L2-error analysis of an isoparametric unfitted finite element method for elliptic interface problems, J. Numer. Math., (2016).
25.
C. Lehrenfeld and A. Reusken, Optimal preconditioners for Nitsche-XFEM discretizations of interface problems, Numer. Math., 135 (2017), pp. 313--332.
26.
R. J. Leveque and Z. Li, The immersed interface method for elliptic equations with discontinuous coefficients and singular sources, SIAM J. Numer. Anal., 31 (1994), pp. 1019--1044.
27.
Z. Li, A fast iterative algorithm for elliptic interface problems, SIAM J. Numer. Anal., 35 (1998), pp. 230--254.
28.
J. Nitsche, Über ein Variationsprinzip zur Lösung von Dirichlet-Problemen bei Verwendung von Teilräumen, die keinen Randbedingungen unterworfen sind, Abh. Math. Semin. Univ. Hamburg, 36, (1971), pp. 9--15.
29.
A. Reusken, Analysis of an extended pressure finite element space for two-phase incompressible flows, Comput. Visual. Sci., 11 (2008), pp. 293--305.
30.
W. L. Wan, Interface preserving coarsening multigrid for elliptic problems with highly discontinuous coefficients, Numer. Linear Algebra Appl., 7 (2000), pp. 727--742.

Information & Authors

Information

Published In

cover image SIAM Journal on Scientific Computing
SIAM Journal on Scientific Computing
Pages: A318 - A342
ISSN (online): 1095-7197

History

Submitted: 26 July 2018
Accepted: 18 November 2019
Published online: 4 February 2020

Keywords

  1. geometric multigrid
  2. unfitted finite elements
  3. interface problem
  4. Nitsche
  5. CutFEM

MSC codes

  1. 65N30
  2. 65F10
  3. 65N22
  4. 65N55

Authors

Affiliations

Funding Information

Deutsche Forschungsgemeinschaft https://doi.org/10.13039/501100001659 : Re 1461/6-1

Metrics & Citations

Metrics

Citations

If you have the appropriate software installed, you can download article citation data to the citation manager of your choice. Simply select your manager software from the list below and click Download.

Cited By

View Options

View options

PDF

View PDF

Media

Figures

Other

Tables

Share

Share

Copy the content Link

Share with email

Email a colleague

Share on social media