Closed-Form Inverses of the Weakly Singular and Hypersingular Operators on Disks



We establish new explicit expressions and variational forms of boundary integral operators that provide the exact inverses of the weakly singular and hypersingular operators for \(-\Delta \) on flat disks. We derive closed-form formulas for their singular kernels. We also show that the inverse of the weakly singular operator can be obtained by composing surface curl operators and the inverse of the hypersingular operator.


Boundary integral operators Screens Exact inverses Projected spherical harmonics 

Mathematics Subject Classification

Primary 45P05 31A10 65R20 Secondary 65N38 


  1. 1.
    McLean, W.: Strongly Elliptic Systems and Boundary Integral Equations. Cambridge University Press, Cambridge (2000)MATHGoogle Scholar
  2. 2.
    Sauter, S., Schwab, C.: Boundary Element Methods, Springer Series in Computational Mathematics, vol. 39. Springer, Heidelberg (2010)Google Scholar
  3. 3.
    Stephan, E.P.: A boundary integral equation method for three-dimensional crack problems in elasticity. Math. Methods Appl. Sci. 8, 609–623 (1986)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Stephan, E.: Boundary integral equations for screen problems in \(\mathbb{R}^3\). Integral Equ. Oper. Theory 10, 236–257 (1987)CrossRefMATHGoogle Scholar
  5. 5.
    Buffa, A., Christiansen, S.H.: The electric field integral equation on Lipschitz screens: definitions and numerical approximation. Numer. Math. 94, 229–267 (2003)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Steinbach, O., Wendland, W.: The construction of some efficient preconditioners in the boundary element method. Adv. Comput. Math. 9, 191–216 (1998)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Hiptmair, R.: Operator preconditioning. Comput. Math. Appl. 52, 699–706 (2006)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Hiptmair, R., Jerez-Hanckes, C., Urzúa-Torres, C.: Mesh-independent operator preconditioning for boundary elements on open curves. SIAM J. Numer. Anal. 52, 2295–2314 (2014)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Lions, J., Magenes, F.: Nonhomogeneous Boundary Value Problems and Applications. Springer, Berlin (1972)CrossRefMATHGoogle Scholar
  10. 10.
    Martin, P.A.: Exact solution of some integral equations over a circular disc. J. Integral Equ. Appl. 18, 39–58 (2006)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Ramaciotti Morales, P.: Theoretical and numerical aspects of wave propagation phenomena in complex domains and applications to remote sensing. PhD Thesis, Université Paris-Saclay (2016)Google Scholar
  12. 12.
    Jerez-Hanckes, C., Nédélec, J.: Explicit variational forms for the inverses of integral logarithmic operators over an interval. SIAM J. Math. Anal. 44, 2666–2694 (2012)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Ramaciotti, P., Nédélec, J.-C.: About some boundary integral operators on the unit disk related to the Laplace equation. SIAM J. Numer. Anal. 55, 1892–1914 (2017)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Steinbach, O.: Numerical Approximation Methods for Elliptic Boundary Value Problems: Finite and Boundary Elements. Springer, New York (2008)CrossRefMATHGoogle Scholar
  15. 15.
    Nédélec, J.-C.: About some operators on the unit disc related to the Laplacian equation. In: Journées Singulières Augmentées (2013).
  16. 16.
    Costabel, M., Dauge, M., Duduchava, R.: Asymptotics without logarithmic terms for crack problems. Commun. Partial Differ. Equ. 28, 869–926 (2003)MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Fabrikant, V.I.: Mixed Boundary Value Problems of Potential Theory and Their Applications in Engineering, Mathematics and its Applications, vol. 68. Kluwer, Dordrecht (1991)MATHGoogle Scholar
  18. 18.
    Li, X.-F., Rong, E.-Q.: Solution of a class of two-dimensional integral equations. J. Comput. Appl. Math. 145, 335–343 (2002)MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    Wolfe, P.: Eigenfunctions of the integral equation for the potential of the charged disk. J. Math. Phys. 12, 1215–1218 (1971)MathSciNetCrossRefMATHGoogle Scholar
  20. 20.
    Krenk, S.: A circular crack under asymmetric loads and some related integral equations. J. Appl. Mech. 46, 821–826 (1979)CrossRefMATHGoogle Scholar
  21. 21.
    NIST Digital library of mathematical functions., Release 1.0.10 of 2015-08-07
  22. 22.
    Bateman, H., Erdélyi, A.: Higher Transcendental Functions. McGraw-Hill, New York (1952)MATHGoogle Scholar
  23. 23.
    Hiptmair, R., Jerez-Hanckes, C., Urzúa-Torres, C.: Optimal operator preconditioning for hypersingular operator over 3D screens. Tech. Rep. 2016-09, Seminar for Applied Mathematics, ETH Zürich, Switzerland (2016).
  24. 24.
    Hiptmair, R., Jerez-Hanckes, C., Urzúa-Torres, C.: Optimal Operator Preconditioning for weakly singular Operator over 3D screens Tech. Rep. 2017-13, Seminar for Applied Mathematics, ETH Zürich, Switzerland (2017).

Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  • R. Hiptmair
    • 1
  • C. Jerez-Hanckes
    • 2
  • C. Urzúa-Torres
    • 1
  1. 1.Seminar for Applied MathematicsETH ZurichZurichSwitzerland
  2. 2.School of EngineeringPontificia Universidad Católica de ChileMaculChile

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