Journal of Scientific Computing

, Volume 70, Issue 2, pp 686–716 | Cite as

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

  • Cheng Wang
  • Pengtao Sun


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.


Fictitious domain method Distributed Lagrange multiplier Fully discrete finite element scheme Subgrid integration Immersed moving interface 

Mathematics Subject Classification

65N30 65R20 



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.


  1. 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)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Bastian, P., Engwer, C.: An unfitted finite element method using discontinuous Galerkin. Int. J. Numer. Methods Eng. 79(12), 1557–1576 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 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)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 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)Google Scholar
  5. 5.
    Chen, Z., Zou, J.: Finite element methods and their convergence for elliptic and parabolic interface problems. Numer. Math. 79(2), 175–202 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Ciarlet, P.G.: Finite Element Method for Elliptic Problems. Society for Industrial and Applied Mathematics, Philadelphia (2002)CrossRefzbMATHGoogle Scholar
  7. 7.
    Cui, S.: Well-posedness of a multidimensional free boundary problem modelling the growth of nonnecrotic tumors. J. Funct. Anal. 245, 1–18 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 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)CrossRefGoogle Scholar
  9. 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)CrossRefzbMATHGoogle Scholar
  10. 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)MathSciNetCrossRefGoogle Scholar
  11. 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)Google Scholar
  12. 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)CrossRefzbMATHGoogle Scholar
  13. 13.
    Girault, V., Raviart, P.A.: Finite Element Methods for Navier-Stokes Equations: Theory and Algorithms, 1st edn. Springer Publishing Company, Incorporated (2011)zbMATHGoogle Scholar
  14. 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)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 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)CrossRefzbMATHGoogle Scholar
  16. 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)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 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)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Gupta, S.C.: The Classical Stefan Problem: Basic Concepts. Modelling and Analysis. Elsevier, Amsterdam (2003)zbMATHGoogle Scholar
  19. 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)Google Scholar
  20. 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)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 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)CrossRefzbMATHGoogle Scholar
  22. 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)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Li, Z.L.: The immersed interface method using a finite element formulation. Appl. Numer. Math. 27(3), 253–267 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 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)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Muntean, A.: Well-posedness of a moving-boundary problem with two moving reaction strips. Nonlinear Anal. Real World Appl. 10, 2541–2557 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 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. 27.
    Parvizian, J., Düster, A., Rank, E.: Finite cell method. Comput. Mech. 41(1), 121–133 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 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)MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Peskin, C.S.: Flow patterns around heart valves: a numerical method. J. Comput. Phys. 10(2), 252–271 (1972)MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Peskin, C.S.: The immersed boundary method. Acta Numer. 11, 479–517 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  31. 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)MathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    Prüss, J., Simonett, G.: Moving Interfaces and Quasilinear Parabolic Evolution Equations. Monographs in Mathematics, vol. 105, 1st edn. Birkhäuser Verlag, Basel (2016)CrossRefzbMATHGoogle Scholar
  33. 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)MathSciNetCrossRefzbMATHGoogle Scholar
  34. 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)CrossRefzbMATHGoogle Scholar
  35. 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)MathSciNetCrossRefzbMATHGoogle Scholar
  36. 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)CrossRefzbMATHGoogle Scholar
  37. 37.
    Yu, Z.: A DLM/FD method for fluid/flexible-body interactions. J. Comput. Phys. 207(1), 1–27 (2005)CrossRefzbMATHGoogle Scholar
  38. 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)MathSciNetCrossRefzbMATHGoogle Scholar
  39. 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)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2016

Authors and Affiliations

  1. 1.Department of MathematicsTongji UniversityShanghaiChina
  2. 2.Department of Mathematical SciencesUniversity of Nevada Las VegasLas VegasUSA

Personalised recommendations