The present work concerns the approximation of the solution map S associated to the parametric Helmholtz boundary value problem, i.e., the map which associates to each (real) wavenumber belonging to a given interval of interest the corresponding solution of the Helmholtz equation. We introduce a least squares rational Padé-type approximation technique applicable to any meromorphic Hilbert space-valued univariate map, and we prove the uniform convergence of the Padé approximation error on any compact subset of the interval of interest that excludes any pole. This general result is then applied to the Helmholtz solution map S, which is proven to be meromorphic in ℂ, with a pole of order one in every (single or multiple) eigenvalue of the Laplace operator with the considered boundary conditions. Numerical tests are provided that confirm the theoretical upper bound on the Padé approximation error for the Helmholtz solution map.
Accepté le :
DOI : 10.1051/m2an/2017050
Mots clés : Hilbert space-valued meromorphic maps, Padé approximants, convergence of Padé approximants, parametric PDEs, Helmholtz equation
@article{M2AN_2018__52_4_1261_0,
author = {Bonizzoni, Francesca and Nobile, Fabio and Perugia, Ilaria},
title = {Convergence analysis of {Pad\'e} approximations for {Helmholtz} frequency response problems},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis },
pages = {1261--1284},
publisher = {EDP-Sciences},
volume = {52},
number = {4},
year = {2018},
doi = {10.1051/m2an/2017050},
mrnumber = {3875286},
zbl = {1411.35079},
language = {en},
url = {http://archive.numdam.org/articles/10.1051/m2an/2017050/}
}
TY - JOUR AU - Bonizzoni, Francesca AU - Nobile, Fabio AU - Perugia, Ilaria TI - Convergence analysis of Padé approximations for Helmholtz frequency response problems JO - ESAIM: Mathematical Modelling and Numerical Analysis PY - 2018 SP - 1261 EP - 1284 VL - 52 IS - 4 PB - EDP-Sciences UR - http://archive.numdam.org/articles/10.1051/m2an/2017050/ DO - 10.1051/m2an/2017050 LA - en ID - M2AN_2018__52_4_1261_0 ER -
%0 Journal Article %A Bonizzoni, Francesca %A Nobile, Fabio %A Perugia, Ilaria %T Convergence analysis of Padé approximations for Helmholtz frequency response problems %J ESAIM: Mathematical Modelling and Numerical Analysis %D 2018 %P 1261-1284 %V 52 %N 4 %I EDP-Sciences %U http://archive.numdam.org/articles/10.1051/m2an/2017050/ %R 10.1051/m2an/2017050 %G en %F M2AN_2018__52_4_1261_0
Bonizzoni, Francesca; Nobile, Fabio; Perugia, Ilaria. Convergence analysis of Padé approximations for Helmholtz frequency response problems. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 52 (2018) no. 4, pp. 1261-1284. doi : 10.1051/m2an/2017050. http://archive.numdam.org/articles/10.1051/m2an/2017050/
[1] and , A primer of nonlinear analysis, volume 34 of Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge (1995). Corrected reprint of the 1993 original. | MR | Zbl
[2] and , Is the pollution effect of the FEM avoidable for the Helmholtz equation considering high wave numbers? SIAM J. Numer. Anal. 34 (1997) 2392–2423. | DOI | MR | Zbl
[3] and , Padé approximants. Cambridge University Press (1996). | DOI | MR | Zbl
[4] , and , A survey of model reduction methods for parametric systems. Preprint MPIMD/13-14, Max Planck Institute Magdeburg (2013). Available from http://www.mpi-magdeburg.mpg.de/preprints/.
[5] , , , , and , Convergence rates for greedy algorithms in reduced basis methods. SIAM J. Math. Anal. 43 (2011) 1457–1472. | DOI | MR | Zbl
[6] , and , Vector-valued meromorphic functions. Archiv der Math. 79 (2002) 353–359. | DOI | MR | Zbl
[7] , , , and , A priori convergence of the greedy algorithm for the parametrized reduced basis method. ESAIM Math. Model. Numer. Anal. 46 (2012) 595–603. | DOI | Numdam | MR | Zbl
[8] , , and , Certified reduced basis methods and output bounds for the harmonic Maxwell’s equations. SIAM J. Sci. Comput. 32 (2010) 970–996. | DOI | MR | Zbl
[9] , The rational Hermite interpolation problem and some related recurrence formulas. Comput. Math. Appl. 2 (1976) 117–123. | DOI | Zbl
[10] , and , Analytic regularity and polynomial approximation of parametric and stochastic elliptic PDEs. Anal. Appl. 09 (2011) 11–47. | DOI | MR | Zbl
[11] , How well can the concept of Padé approximant be generalized to the multivariate case? J. Comput. Appl. Math. 105 (1999) 25–50. | DOI | MR | Zbl
[12] , On rational interpolation to meromorphic functions in several variables. J. Approximation Theory 105 (2000) 211–237. | DOI | MR | Zbl
[13] , On the convergence of certain sequences of rational approximants to meromorphic functions in several variables. J. Approximation Theory 130 (2004) 99–112. | DOI | MR | Zbl
[14] , and , Generalized multivariate Padé approximants. J. Approximation Theory 95 (1998) 203–214. | DOI | MR | Zbl
[15] , and , Review and assessment of interpolatory model order reduction methods for frequency response structural dynamics and acoustics problems. Inter. J. Numer. Methods Eng. 90 (2012) 1636–1662. | DOI | MR | Zbl
[16] , and , An adaptive scheme for a class of interpolatory model reduction methods for frequency response problems. Inter. J. Numer. Methods Eng. 93 (2013) 1109–1124. | DOI | MR | Zbl
[17] and , Continuity of approximation by least-squares multivariate Padé approximants. J. Comput. Appl. Math. 115 (2000) 255–268. | DOI | MR | Zbl
[18] , Extension of vector-valued holomorphic and meromorphic functions. Bull. Belg. Math. Soc. Simon Stevin 12 (2005) 5–21. | DOI | MR | Zbl
[19] and , Rational approximation by an interpolation procedure in several variables. Padé and Rational Approximation (1977) 83–100. | DOI | MR | Zbl
[20] , Perturbation theory for linear operators. Classics in Mathematics. Springer Verlag, Berlin (1995). Reprint of the 1980 edition. | DOI | MR | Zbl
[21] , , and , Generalized reduced basis methods and n-width estimates for the approximation of the solution manifold of parametric PDEs. In Analysis and Numerics of Partial Differential Equations. In volume 4 of Springer INdAM Ser. Springer, Milan (2013) 307–329. | DOI | MR | Zbl
[22] , and , On the approximation of stability factors for general parametrized partial differential equations with a two-level affine decomposition. ESAIM: Math. Model. Numer. Anal. 46 (2012) 1555–1576. | DOI | Numdam | MR | Zbl
[23] , Initial-boundary value problems in mathematical physics. ; John Wiley Sons, Ltd., Chichester (1986). | DOI | MR
[24] and , Convergence analysis of a balancing domain decomposition method for solving a class of indefinite linear systems. Numer. Linear Algebra Appl. 16 (2009) 745–773. | DOI | MR | Zbl
[25] , On Generalized Finite Element Methods. Ph.D. thesis, Univ. Maryland (1995). | MR
[26] , Sur les fractions continues algébriques. Bulletin de la Soc. Math. France 30 (1902) 28–36. | DOI | JFM | Numdam | MR
[27] , Multipoint Padé approximation and orthogonal rational functions. In Nonl. Numer. Methods Rational Approximation, edited by . volume 43 of Math. Appl. Springer Netherlands (1988) 259–270. | DOI | MR | Zbl
[28] and , Analysis and implementation issues for the numerical approximation of parabolic equations with random coefficients. Inter. J. Numer. Methods Eng. 80 (2009) 979–1006. | DOI | MR | Zbl
[29] , , , , and , “Natural norm” a posteriori error estimators for reduced basis approximations. J. Comput. Phys. 217 (2006) 37–62. | DOI | MR | Zbl
[30] , , and , A posteriori error bounds for reduced-basis approximation of parametrized noncoercive and nonlinear elliptic partial differential equations (2003).
[31] , Convergence of Padé approximants in the general case. Rocky Mountain J. Math. 4 (1974) 325–330. | DOI | MR | Zbl
Cité par Sources :
