Skip to main content
SpringerLink
Log in
Book cover

Geometrically Unfitted Finite Element Methods and Applications pp 281–306Cite as

A Space-Time Cut Finite Element Method with Quadrature in Time

  • Conference paper
  • First Online:

Part of the book series: Lecture Notes in Computational Science and Engineering ((LNCSE,volume 121))

Abstract

We consider convection-diffusion problems in time-dependent domains and present a space-time finite element method based on quadrature in time which is simple to implement and avoids remeshing procedures as the domain is moving. The evolving domain is embedded in a domain with fixed mesh and a cut finite element method with continuous elements in space and discontinuous elements in time is proposed. The method allows the evolving geometry to cut through the fixed background mesh arbitrarily and thus avoids remeshing procedures. However, the arbitrary cuts may lead to ill-conditioned algebraic systems. A stabilization term is added to the weak form which guarantees well-conditioned linear systems independently of the position of the geometry relative to the fixed mesh and in addition makes it possible to use quadrature rules in time to approximate the space-time integrals. We review here the space-time cut finite element method presented in Hansbo et al. (Comput. Methods Appl. Mech. Eng. 307: 96–116, 2016) where linear elements are used in both space and time and extend the method to higher order elements for problems on evolving surfaces (or interfaces). We present a new stabilization term which also when higher order elements are used controls the condition number of the linear systems from cut finite element methods on evolving surfaces. The new stabilization combines the consistent ghost penalty stabilization (Burman, C. R. Acad. Sci. Paris, Ser. I 348(21–22):1217–1220, 2010) with a term controlling normal derivatives at the interface.

This is a preview of subscription content, log in via an institution.

Chapter
USD   29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD   129.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD   169.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD   169.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Learn about institutional subscriptions

References

  1. Burman, E.: Ghost penalty. C. R. Acad. Sci. Paris, Ser. I 348(21–22), 1217–1220 (2010)

    Google Scholar 

  2. Burman, E., Hansbo, P.: Fictitious domain finite element methods using cut elements: II. A stabilized Nitsche method. Appl. Numer. Math. 62(4), 328–341 (2012)

    MATH  Google Scholar 

  3. Burman, E., Hansbo, P., Larson, M.G.: A stabilized cut finite element method for partial differential equations on surfaces: the Laplace-Beltrami operator. Comput. Methods Appl. Mech. Eng. 285, 188–207 (2015)

    Article  MathSciNet  Google Scholar 

  4. Burman, E., Hansbo, P., Larson, M.G., Zahedi, S.: Stabilized CutFEM for the convection problem on surfaces. Technical Report, Mathematics, Umeå University, Sweden (2015). ArXiv:1511.02340

    Google Scholar 

  5. Burman, E., Hansbo, P., Larson, M.G., Massing, A.: Cut finite element methods for partial differential equations on embedded manifolds of arbitrary codimensions. Technical Report, Mathematics, Umeå University, Sweden (2016). ArXiv:1610.01660

    Google Scholar 

  6. Burman, E., Hansbo, P., Larson, M.G., Zahedi, S.: Cut finite element methods for coupled bulk-surface problems. Numer. Math. 133(2), 203–231 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  7. Chen, K.Y., Lai, M.C.: A conservative scheme for solving coupled surface-bulk convection–diffusion equations with an application to interfacial flows with soluble surfactant. J. Comput. Phys. 257, 1–18 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  8. Deckelnick, K., Elliott, C.M., Ranner, T.: Unfitted finite element methods using bulk meshes for surface partial differential equations. SIAM J. Numer. Anal. 52(4), 2137–2162 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  9. Fries, T.P.: Towards higher-order XFEM for interfacial flows. Proc. Appl. Math. Mech. 15(1), 507–508 (2015)

    Article  Google Scholar 

  10. Ganesan, S., Tobiska, L.: Arbitrary Lagrangian–Eulerian finite-element method for computation of two-phase flows with soluble surfactants. J. Comput. Phys. 231(9), 3685–3702 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  11. Grande, J.: Eulerian finite element methods for parabolic equations on moving surfaces. SIAM J. Sci. Comput. 36(2), 248–271 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  12. Grande, J., Lehrenfeld, C., Reusken, A.: Analysis of a high order trace finite element method for PDEs on level set surfaces (2016). ArXiv:1611.01100

    Google Scholar 

  13. Hansbo, P., Larson, M.G., Zahedi, S.: A cut finite element method for a Stokes interface problem. Appl. Numer. Math. 85, 90–114 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  14. Hansbo, P., Larson, M.G., Zahedi, S.: Characteristic cut finite element methods for convection–diffusion problems on time dependent surfaces. Comput. Methods Appl. Mech. Eng. 293, 431–461 (2015)

    Article  MathSciNet  Google Scholar 

  15. Hansbo, P., Larson, M.G., Zahedi, S.: A cut finite element method for coupled bulk-surface problems on time-dependent domains. Comput. Methods Appl. Mech. Eng. 307, 96–116 (2016)

    Article  MathSciNet  Google Scholar 

  16. Hou, T.Y., Lowengrub, J.S., Shelley, M.J.: Removing the stiffness from interfacial flows with surface tension. J. Comput. Phys. 114(2), 312–338 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  17. Jamet, P.: Galerkin-type approximations which are discontinuous in time for parabolic equations in a variable domain. SIAM J. Numer. Anal. 15(5), 912–928 (1978)

    Article  MathSciNet  MATH  Google Scholar 

  18. Larson, M.G., Zahedi, S.: Stabilization of higher order cut finite element methods on surfaces. (2017). ArXiv:1710.03343

    Google Scholar 

  19. Lehrenfeld, C.: The Nitsche XFEM-DG space-time method and its implementation in three space dimensions. SIAM J. Sci. Comput. 37(1), A245–A270 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  20. Lehrenfeld, C.: High order unfitted finite element methods on level set domains using isoparametric mappings. Comput. Methods Appl. Mech. Eng. 300, 716–733 (2016)

    Article  MathSciNet  Google Scholar 

  21. Olshanskii, M.A., Reusken, A.: Error analysis of a space-time finite element method for solving PDEs on evolving surfaces. SIAM J. Numer. Anal. 52(4), 2092–2120 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  22. Olshanskii, M.A., Reusken, A., Grande, J.: A finite element method for elliptic equations on surfaces. SIAM J. Numer. Anal. 47(5), 3339–3358 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  23. Olshanskii, M.A., Reusken, A., Xu, X.: An Eulerian space-time finite element method for diffusion problems on evolving surfaces. SIAM J. Numer. Anal. 52(3), 1354–1377 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  24. Osher, S., Sethian, J.A.: Fronts propagating with curvature-dependent speed: Algorithms based on Hamilton-Jacobi formulations. J. Comput. Phys. 79(1), 12–49 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  25. Peskin, C.S.: Numerical analysis of blood flow in the heart. J. Comput. Phys. 25(3), 220–252 (1977)

    Article  MathSciNet  MATH  Google Scholar 

  26. Ravera, F., Ferrari, M., Liggieri, L.: Adsorption and partitioning of surfactants in liquid–liquid systems. Adv. Colloid Interf. Sci. 88(1–2), 129–177 (2000)

    Article  Google Scholar 

  27. Reusken, A.: Analysis of trace finite element methods for surface partial differential equations. IMA J. Numer. Anal. 35(4), 1568–1590 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  28. Saye, R.I.: High-order quadrature methods for implicitly defined surfaces and volumes in hyperrectangles. SIAM J. Sci. Comput. 37(2), A993–A1019 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  29. Sussman, M., Fatemi, E.: An efficient, interface-preserving level set redistancing algorithm and its application to interfacial incompressible fluid flow. SIAM J. Sci. Comput. 20(4), 1165–1191 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  30. Thomée, V.: Galerkin Finite Element Methods for Parabolic Problems. Springer Series in Computational Mathematics, vol. 25, 2nd edn. Springer, Berlin (2006)

    Google Scholar 

Download references

Acknowledgement

This research was supported by the Swedish Research Council Grant No. 2014-4804.

Author information

Authors and Affiliations

  1. Department of Mathematics, KTH Royal Institute of Technology, SE-100 44, Stockholm, Sweden

    Sara Zahedi

Authors
  1. Sara Zahedi
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Sara Zahedi .

Editor information

Editors and Affiliations

  1. Faculté des Sciences, de la Technologie et de la Communication, Université du Luxembourg, Luxembourg

    Stéphane P. A. Bordas

  2. Department of Mathematics, University College London, United Kingdom

    Erik Burman

  3. Department of Mathematics and Mathematical Statistics, Umeå University, Sweden

    Mats G. Larson

  4. Department of Mathematics, University of Houston, USA

    Maxim A. Olshanskii

Rights and permissions

Reprints and permissions

Copyright information

© 2017 Springer International Publishing AG

About this paper

Check for updates. Verify currency and authenticity via CrossMark

Cite this paper

Zahedi, S. (2017). A Space-Time Cut Finite Element Method with Quadrature in Time. 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_9

Download citation

Publish with us

Policies and ethics

Search

Navigation