Foundations of Computational Mathematics

, Volume 17, Issue 3, pp 627–674 | Cite as

Analysis of the Diffuse Domain Method for Second Order Elliptic Boundary Value Problems

  • Martin Burger
  • Ole Løseth Elvetun
  • Matthias SchlottbomEmail author


The diffuse domain method for partial differential equations on complicated geometries recently received strong attention in particular from practitioners, but many fundamental issues in the analysis are still widely open. In this paper, we study the diffuse domain method for approximating second order elliptic boundary value problems posed on bounded domains and show convergence and rates of the approximations generated by the diffuse domain method to the solution of the original second order problem when complemented by Robin, Dirichlet or Neumann conditions. The main idea of the diffuse domain method is to relax these boundary conditions by introducing a family of phase-field functions such that the variational integrals of the original problem are replaced by a weighted average of integrals of perturbed domains. From a functional analytic point of view, the phase-field functions naturally lead to weighted Sobolev spaces for which we present trace and embedding results as well as various types of Poincaré inequalities with constants independent of the domain perturbations. Our convergence analysis is carried out in such spaces as well, but allows to draw conclusions also about unweighted norms applied to restrictions on the original domain. Our convergence results are supported by numerical examples.


Diffuse domain method Weighted Sobolev spaces Domain perturbations Elliptic boundary value problems 

Mathematics Subject Classification

35J20 35J70 46E35 65N85 



MB and MS acknowledge support by ERC via Grant EU FP 7 - ERC Consolidator Grant 615216 LifeInverse. MB acknowledges support by the German Science Foundation DFG via EXC 1003 Cells in Motion Cluster of Excellence, Münster, Germany. OLE acknowledges support by DAAD for his 1 year research stay at WWU Münster.


  1. 1.
    H. Abels, K. F. Lam, and B. Stinner. Analysis of the diffuse domain approach for a bulk-surface coupled PDE system. SIAM J. Math. Anal., 47(5):3687–3725, 2015. doi: 10.1137/15M1009093.
  2. 2.
    R. A. Adams. Sobolev spaces. Academic Press [A subsidiary of Harcourt Brace Jovanovich, Publishers], New York-London, 1975. Pure and Applied Mathematics, Vol. 65.Google Scholar
  3. 3.
    S. Aland, J. Lowengrub, and A. Voigt. Two-phase flow in complex geometries: a diffuse domain approach. CMES Comput. Model. Eng. Sci., 57(1):77–107, 2010.MathSciNetzbMATHGoogle Scholar
  4. 4.
    J. M. Arrieta, A. Rodríguez-Bernal, and J. D. Rossi. The best Sobolev trace constant as limit of the usual Sobolev constant for small strips near the boundary. Proc. Roy. Soc. Edinburgh Sect. A, 138(2):223–237, 2008.MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    I. Babuška. The finite element method with penalty. Math. Comp., 27:221–228, 1973.MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    J. W. Barrett and C. M. Elliott. Finite element approximation of the Dirichlet problem using the boundary penalty method. Numer. Math., 49(4):343–366, 1986.MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    J. W. Barrett and C. M. Elliott. Fitted and unfitted finite-element methods for elliptic equations with smooth interfaces. IMA J. Numer. Anal., 7(3):283–300, 1987.MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    J. W. Barrett and C. M. Elliott. A practical finite element approximation of a semidefinite Neumann problem on a curved domain. Numer. Math., 51(1):23–36, 1987.MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    P. Bastian and C. Engwer. An unfitted finite element method using discontinuous Galerkin. Internat. J. Numer. Methods Engrg., 79(12):1557–1576, 2009.MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    S. Bertoluzza, M. Ismail, and B. Maury. The fat boundary method: semi-discrete scheme and some numerical experiments. In Domain decomposition methods in science and engineering, volume 40 of Lect. Notes Comput. Sci. Eng., pages 513–520. Springer, Berlin, 2005.Google Scholar
  11. 11.
    A. Boulkhemair and A. Chakib. On the uniform Poincaré inequality. Comm. Partial Differential Equations, 32(7-9):1439–1447, 2007.MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    D. Braess. Finite elements. Cambridge University Press, Cambridge, third edition, 2007. Theory, fast solvers, and applications in elasticity theory, Translated from the German by Larry L. Schumaker.Google Scholar
  13. 13.
    F. Brezzi. On the existence, uniqueness and approximation of saddle-point problems arising from Lagrangian multipliers. Rev. Française Automat. Informat. Recherche Opérationnelle Sér. Rouge, 8(R-2):129–151, 1974.Google Scholar
  14. 14.
    M. C. Delfour and J.-P. Zolésio. Shapes and geometries, volume 22 of Advances in Design and Control. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, second edition, 2011. Metrics, analysis, differential calculus, and optimization.Google Scholar
  15. 15.
    H. Egger and M. Schlottbom. Analysis and regularization of problems in diffuse optical tomography. SIAM J. Math. Anal., 42(5):1934–1948, 2010.MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    C. M. Elliott and B. Stinner. Analysis of a diffuse interface approach to an advection diffusion equation on a moving surface. Math. Models Methods Appl. Sci., 19(5):787–802, 2009.MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    C. M. Elliott, B. Stinner, V. Styles, and R. Welford. Numerical computation of advection and diffusion on evolving diffuse interfaces. IMA J. Numer. Anal., 31(3):786–812, 2011.MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    S. Esedo\(\bar{\rm g}\)lu, A. Rätz, and M. Röger. Colliding interfaces in old and new diffuse-interface approximations of Willmore-flow. Commun. Math. Sci., 12(1):125–147, 2014.Google Scholar
  19. 19.
    S. Franz, R. Gärtner, H.-G. Roos, and A. Voigt. A note on the convergence analysis of a diffuse-domain approach. Comput. Methods Appl. Math., 12(2):153–167, 2012.MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    R. Glowinski, T.-W. Pan, and J. Périaux. A fictitious domain method for Dirichlet problem and applications. Comput. Methods Appl. Mech. Engrg., 111(3-4):283–303, 1994.MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    J. B. Greer. An improvement of a recent Eulerian method for solving PDEs on general geometries. J. Sci. Comput., 29(3):321–352, 2006.MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    P. Grisvard. Elliptic Problems in Nonsmooth Domains. Pitman, Boston, 1985.zbMATHGoogle Scholar
  23. 23.
    K. Gröger. A \(W^{1,p}\)-estimate for solutions to mixed boundary value problems for second order elliptic differential equations. Math. Ann., 283(4):679–687, 1989.MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    W. Hackbusch and S. A. Sauter. Composite finite elements for the approximation of PDEs on domains with complicated micro-structures. Numer. Math., 75(4):447–472, 1997.MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    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(47-48):5537–5552, 2002.MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    T. Horiuchi. The imbedding theorems for weighted Sobolev spaces. J. Math. Kyoto Univ., 29(3):365–403, 1989.MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    A. Kufner. Weighted Sobolev spaces. A Wiley-Interscience Publication. John Wiley & Sons, Inc., New York, 1985. Translated from the Czech.Google Scholar
  28. 28.
    K. Y. Lervåg and J. Lowengrub. Analysis of the diffuse-domain method for solving PDEs in complex geometries. Commun. Math. Sci., 13(6):1473–1500, 2015. doi: 10.4310/CMS.2015.v13.n6.a6.
  29. 29.
    R. J. LeVeque and Z. L. Li. The immersed interface method for elliptic equations with discontinuous coefficients and singular sources. SIAM J. Numer. Anal., 31(4):1019–1044, 1994.MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    X. Li, J. Lowengrub, A. Rätz, and A. Voigt. Solving PDEs in complex geometries: a diffuse domain approach. Commun. Math. Sci., 7(1):81–107, 2009.MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    F. Liehr, T. Preusser, M. Rumpf, S. Sauter, and L. O. Schwen. Composite finite elements for 3D image based computing. Comput. Vis. Sci., 12(4):171–188, 2009.MathSciNetCrossRefGoogle Scholar
  32. 32.
    N. G. Meyers. An \({L}^p\)-estimate for the gradient of solutions of second order elliptic divergence equations. Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, 17(3):189–206, 1963.zbMATHGoogle Scholar
  33. 33.
    J. Nečas. Direct methods in the theory of elliptic equations. Springer Monographs in Mathematics. Springer, Heidelberg, 2012. Translated from the 1967 French original by Gerard Tronel and Alois Kufner, Editorial coordination and preface by Šárka Nečasová and a contribution by Christian G. Simader.Google Scholar
  34. 34.
    B. Opic and A. Kufner. Hardy-type inequalities, volume 219 of Pitman Research Notes in Mathematics Series. Longman Scientific & Technical, Harlow, 1990.Google Scholar
  35. 35.
    F. Otto, P. Penzler, A. Rätz, T. Rump, and A. Voigt. A diffuse-interface approximation for step flow in epitaxial growth. Nonlinearity, 17(2):477–491, 2004.MathSciNetCrossRefzbMATHGoogle Scholar
  36. 36.
    J. Parvizian, A. Düster, and E. Rank. Finite cell method: \(h\)-extension for embedded domain problems in solid mechanics. Comput. Mech., 41(1):121–133, 2007.MathSciNetCrossRefzbMATHGoogle Scholar
  37. 37.
    C. S. Peskin. Numerical analysis of blood flow in the heart. J. Computational Phys., 25(3):220–252, 1977.MathSciNetCrossRefzbMATHGoogle Scholar
  38. 38.
    A. Rätz. A new diffuse-interface model for step flow in epitaxial growth. IMA Journal of Applied Mathematics, 2014.Google Scholar
  39. 39.
    A. Rätz, A. Voigt, et al. Pde’s on surfaces—a diffuse interface approach. Communications in Mathematical Sciences, 4(3):575–590, 2006.MathSciNetCrossRefzbMATHGoogle Scholar
  40. 40.
    M. G. Reuter, J. C. Hill, and R. J. Harrison. Solving PDEs in irregular geometries with multiresolution methods I: Embedded Dirichlet boundary conditions. Comput. Phys. Commun., 183(1):1–7, 2012.MathSciNetCrossRefzbMATHGoogle Scholar
  41. 41.
    A. Sarthou, S. Vincent, J. P. Caltagirone, and P. Angot. Eulerian-Lagrangian grid coupling and penalty methods for the simulation of multiphase flows interacting with complex objects. Internat. J. Numer. Methods Fluids, 56(8):1093–1099, 2008.MathSciNetCrossRefzbMATHGoogle Scholar
  42. 42.
    K. E. Teigen, P. Song, J. Lowengrub, and A. Voigt. A diffuse-interface method for two-phase flows with soluble surfactants. J. Comput. Phys., 230(2):375–393, 2011.MathSciNetCrossRefzbMATHGoogle Scholar
  43. 43.
    H. Triebel. Theory of function spaces. III, volume 100 of Monographs in Mathematics. Birkhäuser Verlag, Basel, 2006.Google Scholar
  44. 44.
    Z. Zhang and A. Prosperetti. A second-order method for three-dimensional particle simulation. J. Comput. Phys., 210(1):292–324, 2005.MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© SFoCM 2015

Authors and Affiliations

  • Martin Burger
    • 1
    • 2
  • Ole Løseth Elvetun
    • 3
  • Matthias Schlottbom
    • 1
    Email author
  1. 1.Institute for Computational and Applied MathematicsUniversity of MünsterMünsterGermany
  2. 2.Cells in Motion Cluster of ExcellenceUniversity of MünsterMünsterGermany
  3. 3.Department of Mathematical Sciences and TechnologyNorwegian University of Life SciencesÅsNorway

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