Hostname: page-component-848d4c4894-xm8r8 Total loading time: 0 Render date: 2024-06-28T07:52:58.195Z Has data issue: false hasContentIssue false
  • English
  • Français

Exponential convergence of hp quadrature for integral operators with Gevrey kernels

Published online by Cambridge University Press:  11 October 2010

Alexey Chernov ,
Tobias von Petersdorff  and
Christoph Schwab
Alexey Chernov
Affiliation:
Hausdorff Center for Mathematics and Institute for Numerical Simulation, University of Bonn, Endenicher Allee 60, 53115 Bonn, Germany.
Tobias von Petersdorff
Affiliation:
Department of Mathematics, University of Maryland, College Park, MD 20742, USA.
Christoph Schwab
Affiliation:
Seminar für Angewandte Mathematik, ETH Zürich, 8092 Zürich, Switzerland. schwab@sam.math.ethz.ch
Get access

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.

Keywords

Numerical integrationhypersingular integralsintegral equationsGevrey regularityexponential convergence
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 45 , Issue 3 , May 2011 , pp. 387 - 422
Copyright
© EDP Sciences, SMAI, 2010

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Boutet de, L. Monvel and P. Krée, Pseudo-differential operators and Gevrey classes. Ann. Inst. Fourier (Grenoble) 17 (1967) 295–323. CrossRef
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.
Constantine, G.M. and Savits, T.H., A multivariate Faà di Bruno Formula with applications. Trans. AMS 348 (1996) 503–520. CrossRef
M. Costabel, M. Dauge and S. Nicaise, Corner Singularities and Analytic Regularity for Linear Elliptic Systems. Part I: Smooth domains. HAL Archives (2010), http://hal.archives-ouvertes.fr/docs/00/45/41/17/PDF/CoDaNi_Analytic_Part_I.pdf
DeVore, R.A. and Scott, L.R., Error bounds for Gaussian quadrature and weighted polynomial approximation. SIAM J. Numer. Anal. 21 (1984) 400–412. CrossRef
Duffy, M.G., Quadrature over a pyramid or cube of integrands with a singularity at a vertex. SIAM J. Numer. Anal. 19 (1982) 6. CrossRef
Han, H., The boundary integro-differential equations of three-dimensional Neumann problem in linear elasticity. Numer. Math. 68 (1994) 269–281. CrossRef
G.C. Hsiao and W.L. Wendland, Boundary Integral Equations, Springer Appl. Math. Sci. 164. Springer Verlag (2008).
Hsiao, G.C., Kopp, P. and Wendland, W.L., Galerkin, A collocation method for some integral equations of the first kind. Computing 25 (1980) 89–130. CrossRef
N. Jacob, Pseudodifferential Operators and Markov Processes I: Fourier Analysis and Semigroups. Imperial College Press, London (2001).
R. Kieser, Über einseitige Sprungrelationen und hypersinguläre Operatoren in der Methode der Randelemente. Ph.D. Dissertation, Department of Mathematics, Univ. Stuttgart, Germany (1990).
Maue, A.W., Über die Formulierung eines allgemeinen Diffraktionsproblems mit Hilfe einer Integralgleichung. Zeitschr. f. Physik 126 (1949) 601–618. CrossRef
V.G. Maz'ya, Boundary Integral Equations, in Encyclopedia of Mathematical Sciences 27, Analysis IV, V.G. Maz'ya and S.M. Nikolskii Eds., Springer-Verlag, Berlin (1991) 127–228.
W. McLean, Strongly Elliptic Systems and Boundary Integral Equations. Cambridge Univ. Press, Cambridge (2000).
Nedelec, J.C., Integral Equations with Non-integrable kernels. Integr. Equ. Oper. Theory 5 (1982) 562–572. CrossRef
J.C. Nedelec, Acoustic and Electromagnetic Equations. Springer-Verlag, New York (2001).
N. Reich, Ch. Schwab and Ch. Winter, On Kolmogorov Equations for Anisotropic Multivariate Lévy Processes. Finance Stoch. (2010) DOI: 10.1007/s00780-009-0108-x.
S. Sauter, Über die effiziente Verwendung des Galerkinverfahrens zur Lösung Fredholmscher Integralgleichungen. Ph.D. Thesis, Universität Kiel, Germany (1992).
S. Sauter and C. Schwab, Randelementmethoden. Teubner Publ., Wiesbaden (2004) [English Edition: Boundary Element Methods. Springer Verlag, Berlin-Heidelberg-New York (to appear)].
Schwab, C., Variable order composite quadrature of singular and nearly singular integrals. Computing 53 (1994) 173–194. CrossRef
Schwab, C. and Wendland, W.L., On numerical cubatures of singular surface integrals in boundary element methods. Numer. Math. 62 (1992) 343–369. CrossRef
F. Stenger, Numerical Methods Based on Sinc and Analytic Functions. Springer Series in Computational Mathematics, Springer-Verlag (1993).
Stephan, E.P., 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. CrossRef
Trefethen, L.N., Gauss Quadrature Better, Is than Clenshaw-Curtis? SIAM Rev. 50 (2008) 67–87. CrossRef
Ch. Winter, Wavelet Galerkin schemes for option pricing in multidimensional Lévy models. Ph.D. Thesis No. 18221, ETH Zürich, Switzerland (2009).