Exponential convergence of hp quadrature for integral operators with Gevrey kernels

Alexey Chernov; Tobias von Petersdorff; Christoph Schwab

ESAIM: Mathematical Modelling and Numerical Analysis (2011)

  • Volume: 45, Issue: 3, page 387-422
  • ISSN: 0764-583X

Abstract

top
Galerkin discretizations of integral equations in ℝ d require the evaluation of integrals I = ∫ S ( 1 ) ∫ S ( 2 ) g ( x , y ) d y d x where S(1),S(2) are d-simplices and g has a singularity at x = y. We assume that g is Gevrey smooth for x ≠ y and satisfies bounds for the derivatives which allow algebraic singularities at x = y. This holds for kernel functions commonly occurring in integral equations. We construct a family of quadrature rules 𝒬 N using N function evaluations of g which achieves exponential convergence |I – 𝒬 N | ≤C exp(–rNγ) with constants r, γ > 0.

How to cite

top

Chernov, Alexey, von Petersdorff, Tobias, and Schwab, Christoph. "Exponential convergence of hp quadrature for integral operators with Gevrey kernels." ESAIM: Mathematical Modelling and Numerical Analysis 45.3 (2011): 387-422. <http://eudml.org/doc/276355>.

@article{Chernov2011,
abstract = { Galerkin discretizations of integral equations in $\mathbb\{R\}^\{d\}$ require the evaluation of integrals $I = \int_\{S^\{(1)\}\}\int_\{S^\{(2)\}\}g(x,y)\{\rm d\}y\{\rm d\}x$ where S(1),S(2) are d-simplices and g has a singularity at x = y. We assume that g is Gevrey smooth for x$\ne$y and satisfies bounds for the derivatives which allow algebraic singularities at x = y. This holds for kernel functions commonly occurring in integral equations. We construct a family of quadrature rules $\mathcal\{Q\}_\{N\}$ using N function evaluations of g which achieves exponential convergence |I – $\mathcal\{Q\}_\{N\}$| ≤C exp(–rNγ) with constants r, γ > 0. },
author = {Chernov, Alexey, von Petersdorff, Tobias, Schwab, Christoph},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis},
keywords = {Numerical integration; hypersingular integrals; integral equations; Gevrey regularity; exponential convergence; numerical integration; exponential convergence; numerical examples; Galerkin discretizations; quadrature rules},
language = {eng},
month = {1},
number = {3},
pages = {387-422},
publisher = {EDP Sciences},
title = {Exponential convergence of hp quadrature for integral operators with Gevrey kernels},
url = {http://eudml.org/doc/276355},
volume = {45},
year = {2011},
}

TY - JOUR
AU - Chernov, Alexey
AU - von Petersdorff, Tobias
AU - Schwab, Christoph
TI - Exponential convergence of hp quadrature for integral operators with Gevrey kernels
JO - ESAIM: Mathematical Modelling and Numerical Analysis
DA - 2011/1//
PB - EDP Sciences
VL - 45
IS - 3
SP - 387
EP - 422
AB - Galerkin discretizations of integral equations in $\mathbb{R}^{d}$ require the evaluation of integrals $I = \int_{S^{(1)}}\int_{S^{(2)}}g(x,y){\rm d}y{\rm d}x$ where S(1),S(2) are d-simplices and g has a singularity at x = y. We assume that g is Gevrey smooth for x$\ne$y and satisfies bounds for the derivatives which allow algebraic singularities at x = y. This holds for kernel functions commonly occurring in integral equations. We construct a family of quadrature rules $\mathcal{Q}_{N}$ using N function evaluations of g which achieves exponential convergence |I – $\mathcal{Q}_{N}$| ≤C exp(–rNγ) with constants r, γ > 0.
LA - eng
KW - Numerical integration; hypersingular integrals; integral equations; Gevrey regularity; exponential convergence; numerical integration; exponential convergence; numerical examples; Galerkin discretizations; quadrature rules
UR - http://eudml.org/doc/276355
ER -

References

top
  1. L. Boutet de Monvel and P. Krée, Pseudo-differential operators and Gevrey classes. Ann. Inst. Fourier (Grenoble)17 (1967) 295–323.  
  2. H. Chen and L. Rodino, General Theory of PDE and Gevrey Classes, in General Theory of Partial Differential Equations and Microlocal Analysis, Trieste 1995, Notes Math. Ser.349, Pitman Res., Longman, Harlow (1996) 6–81.  
  3. G.M. Constantine and T.H. Savits, A multivariate Faà di Bruno Formula with applications. Trans. AMS348 (1996) 503–520.  
  4. M. Costabel, M. Dauge and S. Nicaise, Corner Singularities and Analytic Regularity for Linear Elliptic Systems. Part I: Smooth domains. HAL Archives (2010),  URIhttp://hal.archives-ouvertes.fr/docs/00/45/41/17/PDF/CoDaNi_Analytic_Part_I.pdf
  5. R.A. DeVore and L.R. Scott, Error bounds for Gaussian quadrature and weighted polynomial approximation. SIAM J. Numer. Anal.21 (1984) 400–412.  
  6. M.G. Duffy, Quadrature over a pyramid or cube of integrands with a singularity at a vertex. SIAM J. Numer. Anal.19 (1982) 6.  
  7. H. Han, The boundary integro-differential equations of three-dimensional Neumann problem in linear elasticity. Numer. Math.68 (1994) 269–281.  
  8. G.C. Hsiao and W.L. Wendland, Boundary Integral Equations, Springer Appl. Math. Sci.164. Springer Verlag (2008).  
  9. G.C. Hsiao, P. Kopp and W.L. Wendland, A Galerkin collocation method for some integral equations of the first kind. Computing25 (1980) 89–130.  
  10. N. Jacob, Pseudodifferential Operators and Markov Processes I: Fourier Analysis and Semigroups. Imperial College Press, London (2001).  
  11. R. Kieser, Über einseitige Sprungrelationen und hypersinguläre Operatoren in der Methode der Randelemente. Ph.D. Dissertation, Department of Mathematics, Univ. Stuttgart, Germany (1990).  
  12. A.W. Maue, Über die Formulierung eines allgemeinen Diffraktionsproblems mit Hilfe einer Integralgleichung. Zeitschr. f. Physik126 (1949) 601–618.  
  13. V.G. Maz'ya, Boundary Integral Equations, in Encyclopedia of Mathematical Sciences27, Analysis IV, V.G. Maz'ya and S.M. Nikolskii Eds., Springer-Verlag, Berlin (1991) 127–228.  
  14. W. McLean, Strongly Elliptic Systems and Boundary Integral Equations. Cambridge Univ. Press, Cambridge (2000).  
  15. J.C. Nedelec, Integral Equations with Non-integrable kernels. Integr. Equ. Oper. Theory5 (1982) 562–572.  
  16. J.C. Nedelec, Acoustic and Electromagnetic Equations. Springer-Verlag, New York (2001).  
  17. N. Reich, Ch. Schwab and Ch. Winter, On Kolmogorov Equations for Anisotropic Multivariate Lévy Processes. Finance Stoch. (2010) DOI: .  DOI10.1007/s00780-009-0108-x
  18. S. Sauter, Über die effiziente Verwendung des Galerkinverfahrens zur Lösung Fredholmscher Integralgleichungen. Ph.D. Thesis, Universität Kiel, Germany (1992).  
  19. S. Sauter and C. Schwab, Randelementmethoden. Teubner Publ., Wiesbaden (2004) [English Edition: Boundary Element Methods. Springer Verlag, Berlin-Heidelberg-New York (to appear)].  
  20. C. Schwab, Variable order composite quadrature of singular and nearly singular integrals. Computing53 (1994) 173–194.  
  21. C. Schwab and W.L. Wendland, On numerical cubatures of singular surface integrals in boundary element methods. Numer. Math.62 (1992) 343–369.  
  22. F. Stenger, Numerical Methods Based on Sinc and Analytic Functions. Springer Series in Computational Mathematics, Springer-Verlag (1993).  
  23. E.P. Stephan, The hp-version of the boundary element method for solving 2- and 3-dimensional boundary value problems. Comput. Meth. Appl. Mech. Engrg.133 (1996) 183–208.  
  24. L.N. Trefethen, Is Gauss Quadrature Better than Clenshaw-Curtis? SIAM Rev.50 (2008) 67–87.  
  25. Ch. Winter, Wavelet Galerkin schemes for option pricing in multidimensional Lévy models. Ph.D. Thesis No. 18221, ETH Zürich, Switzerland (2009).  

NotesEmbed ?

top

You must be logged in to post comments.