Skip to main content
SpringerLink
Log in

A Fictitious Domain Method with Distributed Lagrange Multiplier for Parabolic Problems With Moving Interfaces

  • Published:
Journal of Scientific Computing Aims and scope Submit manuscript

Abstract

In this paper, we study the fictitious domain method with distributed Lagrange multiplier for the jump-coefficient parabolic problems with moving interfaces. The equivalence between the fictitious domain weak form and the standard weak form of a parabolic interface problem is proved, and the uniform well-posedness of the full discretization of fictitious domain finite element method with distributed Lagrange multiplier is demonstrated. We further analyze the convergence properties for the fully discrete finite element approximation in the norms of \(L^2\), \(H^1\) and a new energy norm. On the other hand, we introduce a subgrid integration technique in order to allow the fictitious domain finite element method to be performed on the triangular meshes without doing any interpolation between the authentic domain and the fictitious domain. Numerical experiments confirm the theoretical results, and show the good performances of the proposed schemes.

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

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12

Similar content being viewed by others

References

  1. Auricchio, F., Boffi, D., Gastaldi, L., Lefieux, A., Reali, A.: On a fictitious domain method with distributed Lagrange multiplier for interface problems. Appl. Numer. Math. 95, 36–50 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  2. Bastian, P., Engwer, C.: An unfitted finite element method using discontinuous Galerkin. Int. J. Numer. Methods Eng. 79(12), 1557–1576 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  3. Bedrossian, J., Von Brecht, J.H., Zhu, S., Sifakis, E., Teran, J.M.: A second order virtual node method for elliptic problems with interfaces and irregular domains. J. Comput. Phys. 229(18), 6405–6426 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  4. Boffi, D., Gastaldi, L., Ruggeri, M.: Mixed formulation for interface problems with distributed Lagrange multiplier. Comput. Math. Appl. 68(12, Part B), 2151–2166 (2014)

  5. Chen, Z., Zou, J.: Finite element methods and their convergence for elliptic and parabolic interface problems. Numer. Math. 79(2), 175–202 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  6. Ciarlet, P.G.: Finite Element Method for Elliptic Problems. Society for Industrial and Applied Mathematics, Philadelphia (2002)

    Book  MATH  Google Scholar 

  7. Cui, S.: Well-posedness of a multidimensional free boundary problem modelling the growth of nonnecrotic tumors. J. Funct. Anal. 245, 1–18 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  8. Dillon, R.H., Fauci, L.J.: An integrative model of internal axoneme mechanics and external fluid dynamics in ciliary beating. J. Theor. Biol. 207, 415–430 (2000)

    Article  Google Scholar 

  9. Donea, J., Giuliani, S., Halleux, J.P.: An arbitrary Lagrangian-Eulerian finite element method for transient dynamic fluid-structure interactions. Comput. Methods Appl. Mech. Eng. 33(1), 689–723 (1982)

    Article  MATH  Google Scholar 

  10. Escherb, J., Zhoua, F., Cui, S.: Well-posedness and stability of a free boundary problem modeling the growth of multi-layer tumors. J. Differ. Equ. 244, 2909–2933 (2008)

    Article  MathSciNet  Google Scholar 

  11. Gander, M., Japhet., C.: Algorithm 932: PANG: software for nonmatching grid projections in 2D and 3D with linear complexity. ACM Trans. Math. Softw. (TOMS), 40(1):Article No. 6, (2013)

  12. Gilmanov, A., Sotiropoulos, F.: A hybrid Cartesian/immersed boundary method for simulating flows with 3d, geometrically complex, moving bodies. J. Comput. Phys. 207(2), 457–492 (2005)

    Article  MATH  Google Scholar 

  13. Girault, V., Raviart, P.A.: Finite Element Methods for Navier-Stokes Equations: Theory and Algorithms, 1st edn. Springer Publishing Company, Incorporated (2011)

    MATH  Google Scholar 

  14. Glowinski, R., Kuznetsov, Y.: Distributed lagrange multipliers based on fictitious domain method for second order elliptic problems. Comput. Methods Appl. Mech. Eng. 196(8), 1498–1506 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  15. Glowinski, R., Pan, T.W., Hesla, T.I., Joseph, D.D.: A distributed Lagrange multiplier/fictitious domain method for particulate flows. Int. J. Multiph. Flow 25(5), 755–794 (1999)

    Article  MATH  Google Scholar 

  16. Glowinski, R., Pan, T.W., Hesla, T.I., Joseph, D.D., Périaux, J.: A fictitious domain approach to the direct numerical simulation of incompressible viscous flow past moving rigid bodies: application to particulate flow. J. Comput. Phys. 169(2), 363–426 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  17. Gong, Y., Li, B., Li, Z.: Immersed-interface finite-element methods for elliptic interface problems with nonhomogeneous jump conditions. SIAM J. Numer. Anal. 46, 472–495 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  18. Gupta, S.C.: The Classical Stefan Problem: Basic Concepts. Modelling and Analysis. Elsevier, Amsterdam (2003)

    MATH  Google Scholar 

  19. Hansbo, A., Hansbo, P.: An unfitted finite element method, based on Nitsches method, for elliptic interface problems. Comput. Methods Appl. Mech. Eng. 191(47C48), 5537–5552 (2002)

  20. He, X., Lin, T., Lin, Y.: Interior penalty bilinear IFE discontinuous Galerkin methods for elliptic equations with discontinuous coefficient. J. Syst. Sci. Complex. 23(3), 467–483 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  21. Hirth, C., Amsden, A.A., Cook, J.: An arbitrary Lagrangian-Eulerian computing method for all flow speeds. J. Comput. Phys. 14(3), 227–253 (1974)

    Article  MATH  Google Scholar 

  22. Huang, J.G., Zou, J.: Some new a priori estimates for second-order elliptic and parabolic interface problems. J. Differ. Equ. 184(2), 570–586 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  23. Li, Z.L.: The immersed interface method using a finite element formulation. Appl. Numer. Math. 27(3), 253–267 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  24. Liu, C., Shen, J.: A phase field model for the mixture of two incompressible fluids and its approximation by a Fourier-spectral method. Phys. D Nonlinear Phenom. 179(34), 211–228 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  25. Muntean, A.: Well-posedness of a moving-boundary problem with two moving reaction strips. Nonlinear Anal. Real World Appl. 10, 2541–2557 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  26. Nicaise, S.: Polygonal Interface Problems. Methoden und Verfahren der Mathematischen Physik (Methods and Procedures in Mathematical Physics), vol. 39. Verlag Peter D. Lang, Frankfurt am Main (1993)

    Google Scholar 

  27. Parvizian, J., Düster, A., Rank, E.: Finite cell method. Comput. Mech. 41(1), 121–133 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  28. Peskin, C.S., McQueen, D.M.: A three-dimensional computational method for blood flow in the heart. 1. immersed elastic fibers in a viscous incompressible fluid. J. Comput. Phys. 81(2), 372–405 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  29. Peskin, C.S.: Flow patterns around heart valves: a numerical method. J. Comput. Phys. 10(2), 252–271 (1972)

    Article  MathSciNet  MATH  Google Scholar 

  30. Peskin, C.S.: The immersed boundary method. Acta Numer. 11, 479–517 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  31. Portegies, J.W., Peletier, M.A.: Well-posedness of a parabolic moving-boundary problem in the setting of Wasserstein gradient flows. Interfaces Free Bound. 12, 121–150 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  32. Prüss, J., Simonett, G.: Moving Interfaces and Quasilinear Parabolic Evolution Equations. Monographs in Mathematics, vol. 105, 1st edn. Birkhäuser Verlag, Basel (2016)

    Book  MATH  Google Scholar 

  33. Sapiro, G., Fedkiw, R.P., Shu, C.W.: Shock capturing, level sets, and PDE based methods in computer vision and image processing: a review of Osher’s contributions. J. Comput. Phys. 185(2), 309–341 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  34. Shi, X., Thien, N.P.: Distributed Lagrange multiplier/fictitious domain method in the framework of lattice Boltzmann method for fluid-structure interactions. J. Comput. Phys. 206(1), 81–94 (2005)

    Article  MATH  Google Scholar 

  35. Sinha, R.K., Deka, B.: On the convergence of finite element method for second order elliptic interface problems. Numer. Funct. Anal. Optim. 27(1), 99–115 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  36. Wachs, A.: Numerical simulation of steady bingham flow through an eccentric annular cross-section by distributed Lagrange multiplier/fictitious domain and augmented Lagrangian methods. J. Non-Newton Fluid Mech. 142, 183–198 (2007)

    Article  MATH  Google Scholar 

  37. Yu, Z.: A DLM/FD method for fluid/flexible-body interactions. J. Comput. Phys. 207(1), 1–27 (2005)

    Article  MATH  Google Scholar 

  38. Zhou, Y.C., Zhao, S., Feig, M., Wei, G.W.: High order matched interface and boundary method for elliptic equations with discontinuous coefficients and singular sources. J. Comput. Phys. 213(1), 1–30 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  39. Zhu, L.D., Peskin, C.S.: Simulation of a flapping flexible filament in a flowing soap film by the immersed boundary method. J. Comput. Phys. 179(2), 452–468 (2002)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgments

P. Sun was partially supported by NSF Grant DMS-1418806 and UNLV Faculty Opportunity Award (2013–2015); C. Wang was supported by UNLV Faculty Opportunity Award during his visit at UNLV in 2014.

Author information

Authors and Affiliations

  1. Department of Mathematics, Tongji University, Shanghai, 200092, China

    Cheng Wang

  2. Department of Mathematical Sciences, University of Nevada Las Vegas, 4505 Maryland Parkway, Las Vegas, NV, 89154, USA

    Pengtao Sun

Authors
  1. Cheng Wang
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Pengtao Sun
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Pengtao Sun.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Wang, C., Sun, P. A Fictitious Domain Method with Distributed Lagrange Multiplier for Parabolic Problems With Moving Interfaces. J Sci Comput 70, 686–716 (2017). https://doi.org/10.1007/s10915-016-0262-1

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10915-016-0262-1

Keywords

Mathematics Subject Classification

Search

Navigation