SPECTRAL ELEMENT APPROXIMATION OF FREDHOLM INTEGRAL EIGENVALUE PROBLEMS

SPECTRAL ELEMENT APPROXIMATION OF FREDHOLM INTEGRAL EIGENVALUE PROBLEMS

Oliveira, S. P.

Full Article:

The Karhunen-Loève expansion of a Gaussian process, a common tool on finite element methods for differential equations with stochastic coefficients, is based on the spectral decomposition of its covariance function. The eigenpairs of the covariance are expressed as a Fredholm integral equation of second kind, which can be readily approximated with piecewise-constant finite elements. In this work, the spectral element method with Gauss- Lobatto-Legendre (GLL) collocation points is employed to approximate this eigenvalue problem. Similarly to piecewise-constant finite elements, this approach is simple to implement and does not lead to generalized discrete eigenvalue problems (considering that the numerical integration is also performed with GLL points), with the additional advantage of providing high-order approximations of the eigenfunctions. Numerical experiments involving covariance functions in one- and two-dimensional domains illustrate the effectiveness of this approach.

The Karhunen-Loève expansion of a Gaussian process, a common tool on finite element methods for differential equations with stochastic coefficients, is based on the spectral decomposition of its covariance function. The eigenpairs of the covariance are expressed as a Fredholm integral equation of second kind, which can be readily approximated with piecewise-constant finite elements. In this work, the spectral element method with Gauss- Lobatto-Legendre (GLL) collocation points is employed to approximate this eigenvalue problem. Similarly to piecewise-constant finite elements, this approach is simple to implement and does not lead to generalized discrete eigenvalue problems (considering that the numerical integration is also performed with GLL points), with the additional advantage of providing high-order approximations of the eigenfunctions. Numerical experiments involving covariance functions in one- and two-dimensional domains illustrate the effectiveness of this approach.

Palavras-chave:

DOI: 10.5151/meceng-wccm2012-19354

Referências bibliográficas

• [1] Azevedo J. S. ,MuradM. A., BorgesM. R., Oliveira S. P. “A spacetime multiscale method for computing statistical moments in strongly heterogeneous poroelastic media of evolving scales”. Int. J. Numer. Meth. Engrg., 2012, in press.
• [2] Azevedo J. S. , Oliveira S. P. “A numerical comparison between quasi-Monte Carlo and sparse grid stochastic collocation methods”. Commun. Comput. Phys. 12, 1051-1069, 2012.
• [3] Castrilln-Cands J. H. and Amaratunga K. “Fast Estimation of Continuous KarhunenLoève Eigenfunctions Using Waveletss”. IEEE Trans. Signal Process. 50, 205-228, 2002.
• [4] Frauenfelder P., Schwab C. and Todor R. “Finite elements for elliptic problems with stochastic coefficients”. Comput. Meth. Appl. Mech. Engrg. 194, 205-228, 2005.
• [5] Komatitsch D and Tromp J. “Introduction to the spectral-element method for 3-D seismic wave propagation”. Geophys. J. Int. 139, 806-822, 1999.
• [6] Oliveira S. P. and Seriani G. “Effect of Element Distortion on the Numerical Dispersion of Spectral Element Methods”. Commun. Comput. Phys. 9, 937-958, 2011.
• [7] Tang T., Xu X and Cheng J. “On spectral methods for Volterra integral equations and the convergence analysis”. J. Comput. Math. 26, 825837, 2008.

Como citar:

Oliveira, S. P.; "SPECTRAL ELEMENT APPROXIMATION OF FREDHOLM INTEGRAL EIGENVALUE PROBLEMS", p-3358-3363. In: In Proceedings of the 10th World Congress on Computational Mechanics [= Blucher Mechanical Engineering Proceedings, v. 1, n. 1]. São Paulo: Blucher, 2014.
ISSN 23580828, DOI 10.5151/meceng-wccm2012-19354