Remember Me
Or use your Academic/Social account:


Or use your Academic/Social account:


You have just completed your registration at OpenAire.

Before you can login to the site, you will need to activate your account. An e-mail will be sent to you with the proper instructions.


Please note that this site is currently undergoing Beta testing.
Any new content you create is not guaranteed to be present to the final version of the site upon release.

Thank you for your patience,
OpenAire Dev Team.

Close This Message


Verify Password:
Verify E-mail:
*All Fields Are Required.
Please Verify You Are Human:
fbtwitterlinkedinvimeoflicker grey 14rssslideshare1
Moiola A.; Hiptmair R.; Perugia I. (2011)
Journal: ZAMP
Languages: English
Types: Article

Classified by OpenAIRE into

arxiv: Mathematics::Complex Variables
Vekua operators map harmonic functions defined on domain in \mathbb R2R2 to solutions of elliptic partial differential equations on the same domain and vice versa. In this paper, following the original work of I. Vekua (Ilja Vekua (1907–1977), Soviet-Georgian mathematician), we define Vekua operators in the case of the Helmholtz equation in a completely explicit fashion, in any space dimension N ≥ 2. We prove (i) that they actually transform harmonic functions and Helmholtz solutions into each other; (ii) that they are inverse to each other; and (iii) that they are continuous in any Sobolev norm in star-shaped Lipschitz domains. Finally, we define and compute the generalized harmonic polynomials as the Vekua transforms of harmonic polynomials. These results are instrumental in proving approximation estimates for solutions of the Helmholtz equation in spaces of circular, spherical, and plane waves. \ud
  • The results below are discovered through our pilot algorithms. Let us know how we are doing!

    • [1] R. Adams, Sobolev spaces, Pure and Applied Mathematics, Academic Press, 1975.
    • [2] J. Avery, Hyperspherical Harmonics: applications in quantum theory, Reidel texts in the mathematical sciences, Kluwer Academic Publishers, 1989.
    • [3] S. Axler, P. Bourdon, and W. Ramey, Harmonic function theory, Graduate Texts in Mathematics, Springer-Verlag, New York, 2001.
    • [4] I. Babuˇska and J. Melenk, The partition of unity method, Int. J. Numer. Methods Eng., 40 (1997), pp. 727-758.
    • [5] A. H. Barnett and T. Betcke, Stability and convergence of the method of fundamental solutions for Helmholtz problems on analytic domains, J. Comput. Phys., 227 (2008), pp. 7003-7026.
    • [6] , An exponentially convergent nonpolynomial finite element method for time-harmonic scattering from polygons, SIAM J. Sci. Comput., 32 (2010), pp. 1417-1441.
    • [7] T. Betcke, Numerical computation of eigenfunctions of planar regions, PhD thesis, University of Oxford, 2005.
    • [8] O. Cessenat and B. Despr´es, Application of an ultra weak variational formulation of elliptic PDEs to the two-dimensional Helmholtz equation, SIAM J. Numer. Anal., 35 (1998), pp. 255-299.
    • [9] A. Charalambopoulos and G. Dassios, On the Vekua pair in spheroidal geometry and its role in solving boundary value problems, Appl. Anal., 81 (2002), pp. 85-113.
    • [10] D. Colton, Bergman operators for elliptic equations in three independent variables, Bull. Amer. Math. Soc., 77 (1971), pp. 752-756.
    • [11] , Integral operators for elliptic equations in three independent variables. I, Applicable Anal., 4 (1974/75), pp. 77-95.
    • [12] , Integral operators for elliptic equations in three independent variables. II, Applicable Anal., 4 (1974/75), pp. 283-295.
    • [13] D. Colton and R. Kress, Inverse Acoustic and Electromagnetic Scattering Theory, vol. 93 of Applied Mathematical Sciences, Springer, Heidelberg, 2nd ed., 1998.
    • [14] R. Courant and D. Hilbert, Methods of mathematical physics. Volume II: partial differential equations, Interscience Publishers, 1962.
    • [15] S. C. Eisenstat, On the rate of convergence of the Bergman-Vekua method for the numerical solution of elliptic boundary value problems, SIAM J. Numer. Anal., 11 (1974), pp. 654-680.
    • [16] C. Farhat, I.Harari, and L. Franca, The discontinuous enrichment method, Comput. Methods Appl. Mech. Eng., 190 (2001), pp. 6455-6479.
    • [17] D. Gilbarg and N. Trudinger, Elliptic Partial Differential Equations of Second Order, Classics in Mathematics, Springer-Verlag, 2nd ed., 1983.
    • [18] R. P. Gilbert and C. Y. Lo, On the approximation of solutions of elliptic partial differential equations in two and three dimensions, SIAM J. Math. Anal., 2 (1971), pp. 17-30.
    • [19] C. Gittelson, R. Hiptmair, and I. Perugia, Plane wave discontinuous Galerkin methods: analysis of the h-version, M2AN Math. Model. Numer. Anal., 43 (2009), pp. 297-332.
    • [20] P. Henrici, A survey of I. N. Vekua's theory of elliptic partial differential equations with analytic coefficients, Z. Angew. Math. Phys., 8 (1957), pp. 169-202.
    • [21] R. Hiptmair, A. Moiola, and I. Perugia, Approximation by plane waves, Preprint 2009-27, SAM Report, ETH Zu¨rich, Switzerland, 2009.
    • [22] , Plane wave discontinuous Galerkin methods for the 2D Helmholtz equation: analysis of the pversion, SIAM J. Numer. Anal., 49 (2011), pp. 264-284.
    • [23] N. T. Hop, Some extensions of I. N. Vekua's method in higher dimensions. I. Integral representation of solutions of an elliptic system. Properties of solutions, Math. Nachr., 130 (1987), pp. 17-34.
    • [24] , Some extensions of I. N. Vekua's method in higher dimensions. II. Boundary value problems, Math. Nachr., 130 (1987), pp. 35-46.
    • [25] M. Ikehata, Probe method and a Carleman function, Inverse Problems, 23 (2007), pp. 1871-1894.
    • [26] N. Lebedev, Special functions and their applications, Prentice-Hall, Englewood Cliffs, N.J., 1965.
    • [27] J. Melenk, On Generalized Finite Element Methods, PhD thesis, University of Maryland, 1995.
    • [28] , Operator adapted spectral element methods I: harmonic and generalized harmonic polynomials, Numer. Math., 84 (1999), pp. 35-69.
    • [29] J. Melenk and I. Babuˇska, Approximation with harmonic and generalized harmonic polinomials in the partition of unit method, Comput. Assist. Mech. Eng. Sci., 4 (1997), pp. 607-632.
    • [30] A. Moiola, Approximation properties of plane wave spaces and application to the analysis of the plane wave discontinuous Galerkin method, Tech. Rep. 2009-06, SAM Report, ETH Zu¨rich, Switzerland, 2009.
    • [31] A. Moiola, R. Hiptmair, and I. Perugia, Plane wave approximation of homogeneous Helmholtz solutions, Z. Angew. Math. Phys., to appear (2011).
    • [32] P. Monk and D. Wang, A least squares method for the Helmholtz equation, Comput. Methods Appl. Mech. Eng., 175 (1999), pp. 121-136.
    • [33] C. Mu¨ller, Spherical harmonics, Lecture notes in Mathematics, Springer-Verlag, 1966.
    • [34] I. Vekua, Solutions of the equation Δu + λ2u = 0, Soobshch. Akad. Nauk Gruz. SSR, 3 (1942), pp. 307- 314.
    • [37] G. Watson, Theory of Bessel functions, Cambridge University Press, 1966.
    • [38] N. Weck, Approximation by Herglotz wave functions, Math. Methods Appl. Sci., 27 (2004), pp. 155-162.
  • No related research data.
  • Discovered through pilot similarity algorithms. Send us your feedback.

Share - Bookmark

Published in

Funded by projects

Cite this article