«Approximate approximations» and the cubature of potentials

Gunther Schmidt

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni (1995)

  • Volume: 6, Issue: 3, page 161-184
  • ISSN: 1120-6330

Abstract

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The paper discusses new cubature formulas for classical integral operators of mathematical physics based on the «approximate approximation» of the density with Gaussian and related functions. We derive formulas for the cubature of harmonic, elastic and diffraction potentials approximating with high order in some range relevant for numerical computations. We prove error estimates and provide numerical results for the Newton potential.

How to cite

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Schmidt, Gunther. "«Approximate approximations» and the cubature of potentials." Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni 6.3 (1995): 161-184. <http://eudml.org/doc/244089>.

@article{Schmidt1995,
abstract = {The paper discusses new cubature formulas for classical integral operators of mathematical physics based on the «approximate approximation» of the density with Gaussian and related functions. We derive formulas for the cubature of harmonic, elastic and diffraction potentials approximating with high order in some range relevant for numerical computations. We prove error estimates and provide numerical results for the Newton potential.},
author = {Schmidt, Gunther},
journal = {Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni},
keywords = {Multivariate quasi-interpolation; Cubature formulas; Harmonic, elastic, diffraction potentials; approximate approximations; cubature of potentials; error analysis; singular integrals; Laguerre polynomials; Gaussian weight; error bounds; numerical results; harmonic potential},
language = {eng},
month = {10},
number = {3},
pages = {161-184},
publisher = {Accademia Nazionale dei Lincei},
title = {«Approximate approximations» and the cubature of potentials},
url = {http://eudml.org/doc/244089},
volume = {6},
year = {1995},
}

TY - JOUR
AU - Schmidt, Gunther
TI - «Approximate approximations» and the cubature of potentials
JO - Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni
DA - 1995/10//
PB - Accademia Nazionale dei Lincei
VL - 6
IS - 3
SP - 161
EP - 184
AB - The paper discusses new cubature formulas for classical integral operators of mathematical physics based on the «approximate approximation» of the density with Gaussian and related functions. We derive formulas for the cubature of harmonic, elastic and diffraction potentials approximating with high order in some range relevant for numerical computations. We prove error estimates and provide numerical results for the Newton potential.
LA - eng
KW - Multivariate quasi-interpolation; Cubature formulas; Harmonic, elastic, diffraction potentials; approximate approximations; cubature of potentials; error analysis; singular integrals; Laguerre polynomials; Gaussian weight; error bounds; numerical results; harmonic potential
UR - http://eudml.org/doc/244089
ER -

References

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  1. ABRAMOWTTZ, M. - STEGUN, I. A., Handbook of Mathematical Functions. Dover, New York1970. 
  2. BATEMAN, H. - ERDÉLYI, A., Tables of Integral Transforms. Vol. II. Mc Graw-Hill, New York1954. Zbl0058.34103MR65685
  3. BEATSON, R. K. - LIGHT, W. A., Quasi-interpolation in the absence of polynomial reproduction. In: D. BRAESS - L. L. SCHUMAKER (eds.), Numerical Methods of Approximation Theory. Birkhäuser, vol. 9, Basel1992, 21-39. Zbl0832.41011MR1269353
  4. MAZ'YA, V., A New Approximation Method and its Applications to the Calculation of Volume Potentials. Boundary Point Method. 3. DFG-Kolloquium des DFG-Forschungsschwerpunktes «Randelementmethoden», SchloßReisensburg, 1991, 18. 
  5. MAZ'YA, V., Approximate approximations. In: J. R. WHITEMAN (éd.), The Mathematics of Finite Elements and Applications. Highlights 1993. Wiley & Sons, Chichester, 77-104. Zbl0827.65083MR1291219
  6. MAZ'YA, V. - KARLIN, V., Semi-analytic time marching algorithms for semi-linear parabolic equations. BIT, 34, 1994, 129-147. Zbl0821.65063MR1429694DOI10.1007/BF01935022
  7. MAZ'YA, V. - SCHMIDT, G., On Approximate Approximations. Preprint LiTH-MAT-R-94-12, Linköping Univ., Dept. of Math., Linköping1994. Zbl1120.41013
  8. MAZ'YA, V. - SCHMIDT, G., On Approximate Approximations using Gaussian Kernels. WIAS Preprint 111, Berlin1994. Zbl0838.65005MR1367394DOI10.1093/imanum/16.1.13
  9. STEIN, E. M., Singular Integrals and Differentiability Properties of Functions. University Press, Princeton1970. Zbl0207.13501MR290095
  10. STEIN, E. M. - WEISS, G., Introduction to Fourier Analysis on Euclidean Spaces. University Press, Princeton1971. Zbl1026.42001MR304972

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