Computational Mechanics

, Volume 51, Issue 4, pp 399–419 | Cite as

Classical FEM-BEM coupling methods: nonlinearities, well-posedness, and adaptivity

  • Markus Aurada
  • Michael Feischl
  • Thomas Führer
  • Michael Karkulik
  • Jens Markus Melenk
  • Dirk Praetorius
Original Paper


We consider a (possibly) nonlinear interface problem in 2D and 3D, which is solved by use of various adaptive FEM-BEM coupling strategies, namely the Johnson–Nédélec coupling, the Bielak–MacCamy coupling, and Costabel’s symmetric coupling. We provide a framework to prove that the continuous as well as the discrete Galerkin solutions of these coupling methods additionally solve an appropriate operator equation with a Lipschitz continuous and strongly monotone operator. Therefore, the original coupling formulations are well-defined, and the Galerkin solutions are quasi-optimal in the sense of a Céa-type lemma. For the respective Galerkin discretizations with lowest-order polynomials, we provide reliable residual-based error estimators. Together with an estimator reduction property, we prove convergence of the adaptive FEM-BEM coupling methods. A key point for the proof of the estimator reduction are novel inverse-type estimates for the involved boundary integral operators which are advertized. Numerical experiments conclude the work and compare performance and effectivity of the three adaptive coupling procedures in the presence of generic singularities.


Nonlinear FEM-BEM coupling Adaptivity 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Ainsworth M, Oden JT (2000) A posteriori error estimation in finite element analysis. Wiley-Interscience, Wiley, New-YorkzbMATHCrossRefGoogle Scholar
  2. 2.
    Aurada M, Ebner M, Feischl M, Ferraz-Leite S, Goldenits P, Karkulik M, Mayr M, Praetorius D (2011) HILBERT: a Matlab implementation of adaptive 2D-BEM. ASC Report 24/2011, Institute for Analysis and Scientific Computing, Vienna University of Technology, Wien, software download at
  3. 3.
    Aurada M, Ferraz-Leite S, Praetorius D (2012) Estimator reduction and convergence of adaptive BEM. Appl Numer Math 62: 787–801MathSciNetzbMATHCrossRefGoogle Scholar
  4. 4.
    Aurada M, Feischl M, Führer T, Karkulik M, Melenk JM, Praetorius D (2012) Inverse estimates for elliptic integral operators and application to the adaptive coupling of FEM and BEM (preprint)Google Scholar
  5. 5.
    Aurada M, Feischl M, Karkulik M, Praetorius D (2012) A posteriori error estimates for the Johnson–Nédélec FEM-BEM coupling. Eng Anal Bound Elem 36: 255–266MathSciNetzbMATHCrossRefGoogle Scholar
  6. 6.
    Aurada M, Feischl M, Praetorius D (2012) Convergence of some adaptive FEM-BEM coupling for elliptic but possibly nonlinear interface problems. Math Model Numer Anal 46: 1147–1173MathSciNetCrossRefGoogle Scholar
  7. 7.
    Aurada M, Karkulik M, Praetorius D (2012) Simple error estimates for hypersingular integral equations in adaptive 3D-BEM (in progress)Google Scholar
  8. 8.
    Bielak J, MacCamy RC (1983/1984) An exterior interface problem in two-dimensional elastodynamics. Quart Appl Math 41:143–159Google Scholar
  9. 9.
    Carstensen C (1996) A posteriori error estimate for the symmetric coupling of finite elements and boundary elements. Computing 57: 301–322MathSciNetzbMATHCrossRefGoogle Scholar
  10. 10.
    Carstensen C, Funken SA, Stephan EP (1997) On the adaptive coupling of FEM and BEM in 2-d-elasticity. Numer Math 77: 187–221MathSciNetzbMATHCrossRefGoogle Scholar
  11. 11.
    Carstensen C, Maischak M, Stephan EP (2001) A posteriori error estimate and h-adaptive algorithm on surfaces for Symm’s integral equation. Numer Math 90: 197–213MathSciNetzbMATHCrossRefGoogle Scholar
  12. 12.
    Carstensen C, Stephan E (1995) Adaptive coupling of boundary elements and finite elements. Math Model Numer Anal 29: 779–817MathSciNetzbMATHGoogle Scholar
  13. 13.
    Cascon M, Kreuzer C, Nochetto R, Siebert K (2008) Quasi-optimal convergence rate for an adaptive finite element method. SIAM J Numer Anal 46: 2524–2550MathSciNetzbMATHCrossRefGoogle Scholar
  14. 14.
    Costabel MA (1988) symmetric method for the coupling of finite elements and boundary elements. In: Whiteman J (ed) The mathematics of finite elements and applications IV, MAFELAP 1987, Academic Press, London, pp 281–288Google Scholar
  15. 15.
    Costabel M, Ervin VJ, Stephan EP (1991) Experimental convergence rates for various couplings of boundary and finite elements. Math Comput Model 15: 93–102MathSciNetzbMATHCrossRefGoogle Scholar
  16. 16.
    Costabel M, Stephan EP (1990) Coupling of finite and boundary element methods for an elastoplastic interface problem. SIAM J Numer Anal 27: 1212–1226MathSciNetzbMATHCrossRefGoogle Scholar
  17. 17.
    Feischl M, Karkulik M, Melenk JM, Praetorius D (2011) Quasi-optimal convergence rate for an adaptive boundary element method. ASC Report 28/2011, Institute for Analysis and Scientific Computing, Vienna University of Technology, WienGoogle Scholar
  18. 18.
    Gatica G, Hsiao G (1995) Boundary-field equation methods for a class of nonlinear problems. Longman, HarlowzbMATHGoogle Scholar
  19. 19.
    Gatica G, Hsiao G, Sayas FJ (2012) Relaxing the hypotheses of Bielak–MacCamy’s BEM-FEM coupling. Numer Math 120: 465–487MathSciNetzbMATHCrossRefGoogle Scholar
  20. 20.
    Graham I, Hackbusch W, Sauter S (2005) Finite elements on degenerate meshes: inverse-type inequalities and applications. IMA J Numer Anal 25: 379–407MathSciNetzbMATHCrossRefGoogle Scholar
  21. 21.
    Hsiao G, Wendland W (2008) Boundary integral equations. Applied mathematical sciences 164. Springer, BerlinCrossRefGoogle Scholar
  22. 22.
    Johnson C, Nédélec JC (1980) On the coupling of boundary integral and finite element methods. Math Comp 35: 1063–1079MathSciNetzbMATHCrossRefGoogle Scholar
  23. 23.
    Leydecker F, Maischak M, Stephan EP, Teltscher M (2010) Adaptive FE-BE coupling for an electromagnetic problem in \({\mathbb {R}^3}\) : a residual error estimator. Math Methods Appl Sci 33: 2162–2186MathSciNetzbMATHCrossRefGoogle Scholar
  24. 24.
    McLean W (2000) Strongly elliptic systems and boundary integral equations. Cambridge University Press, CambridgezbMATHGoogle Scholar
  25. 25.
    Sauter S, Schwab C (2011) Boundary element methods. Springer, BerlinzbMATHCrossRefGoogle Scholar
  26. 26.
    Sayas FJ (2009) The validity of Johnson–Nédélec’s BEM-FEM coupling on polygonal interfaces. SIAM J Numer Anal 47: 3451–3463MathSciNetzbMATHCrossRefGoogle Scholar
  27. 27.
    Scott LR, Zhang S (1990) Finite element interpolation of nonsmooth functions satisfying boundary conditions. Math Comp 54: 483–493MathSciNetzbMATHCrossRefGoogle Scholar
  28. 28.
    Steinbach O (2008) Numerical approximation methods for elliptic boundary value problems: Finite and boundary elements. Springer, New YorkzbMATHCrossRefGoogle Scholar
  29. 29.
    Steinbach O (2011) A note on the stable one-equation coupling of finite and boundary elements. SIAM J Numer Anal 49: 1521–1531MathSciNetzbMATHCrossRefGoogle Scholar
  30. 30.
    Stephan EP, Maischak M (2005) A posteriori error estimates for fem-bem couplings of three-dimensional electromagnetic problems. Methods Appl Mech Eng 1994: 441–452MathSciNetCrossRefGoogle Scholar
  31. 31.
    Stevenson R (2008) The completion of locally refined simplicial partitions created by bisection. Math Comp 77: 227–241MathSciNetzbMATHCrossRefGoogle Scholar
  32. 32.
    Zeidler E (1990) Nonlinear functional analysis and its applications, part II/B. Springer, New YorkCrossRefGoogle Scholar
  33. 33.
    Zienkiewicz OC, Kelly DW, Bettess P (1979) Marriage la mode: the best of both worlds (finite elements and boundary integrals). Energy methods in finite element analysis. Wiley, ChichesterGoogle Scholar

Copyright information

© Springer-Verlag 2012

Authors and Affiliations

  • Markus Aurada
    • 1
  • Michael Feischl
    • 1
  • Thomas Führer
    • 1
  • Michael Karkulik
    • 1
  • Jens Markus Melenk
    • 1
  • Dirk Praetorius
    • 1
  1. 1.Vienna University of TechnologyInstitute for Analysis and Scientific ComputingViennaAustria

Personalised recommendations