×

On condition numbers in \(hp\)-FEM with Gauss-Lobatto-based shape functions. (English) Zbl 0997.65128

The aim of this paper is to study the condition number in the \(hp\)-version of the finite element method (FEM) with Gauss-Lobatto-based shape functions. Here the sharp bounds on the condition number of the stiffness matrices arising in \(h/p\) spectral discretizations for two-dimensional elliptic problems are given. Two types of shape functions that are based on Lagrange interpolation polynomials in the Gauss-Lobatto points are considered. These shape functions result in condition numbers \(O(p)\) and \(O(p\ln p)\) for the condensed stiffness matrices, where \(p\) is the polynomial degree employed. Locally refined meshes are analyzed:
Main result: For the discretization of Dirichlet problems on meshes that are refined geometrically toward singularities, the conditioning of the stiffness matrix is shown to be independent of the number of layers of geometric refinement. The bounds for the condition number on quadrilateral elements are given. Finally, the proof of several technical results is proposed.

MSC:

65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
65N35 Spectral, collocation and related methods for boundary value problems involving PDEs
35J25 Boundary value problems for second-order elliptic equations
65F35 Numerical computation of matrix norms, conditioning, scaling
65F10 Iterative numerical methods for linear systems
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Adams, R. A., Sobolev Spaces (1975), Academic Press: Academic Press New York · Zbl 0186.19101
[2] M. Azaiez, M. Dauge, Y. Maday, Méthodes spectrales et des éléments spectraux, Technical Report 94-17, IRMAR, Université de Rennes 1, Rennes, 1994.; M. Azaiez, M. Dauge, Y. Maday, Méthodes spectrales et des éléments spectraux, Technical Report 94-17, IRMAR, Université de Rennes 1, Rennes, 1994.
[3] Babuška, I.; Craig, A.; Mandel, J.; Pitkäranta, J., Efficient preconditioning for the \(p\) version finite element method in two dimensions, SIAM J. Numer. Anal., 28, 3, 624-661 (1991) · Zbl 0754.65083
[4] Babuška, I.; Guo, B., The hp version of the finite element method for domains with curved boundaries, SIAM J. Numer. Anal., 25, 837-861 (1998) · Zbl 0655.65124
[5] Babuška, I.; Kellogg, R. B.; Pitkäranta, J., Direct and inverse error estimates for finite elements with mesh refinements, Numer. Math., 33, 447-471 (1979) · Zbl 0423.65057
[6] Bank, R.; Scott, R., On the conditioning of finite element equations with highly refined meshes, SIAM J. Numer. Anal., 26, 1383-1394 (1989) · Zbl 0688.65062
[7] Bernardi, C.; Maday, Y., Approximations spectrales de problèmes aux limites elliptiques, Mathématiques & Applications (1992), Springer: Springer Berlin · Zbl 0773.47032
[8] Canuto, C.; Hussaini, M. Y.; Quarteroni, A.; Zhang, T. A., Spectral Methods in Fluid Dynamics (1986), Springer: Springer Berlin
[9] Casarin, M. A., Diagonal edge preconditioning in \(p\)-version and spectral element methods, SIAM J. Sci. Statist. Comput., 18, 2, 610-620 (1997) · Zbl 0939.65070
[10] G. Golub, C. van Loan, Matrix Computations. Johns Hopkins, 3rd edition, 1996.; G. Golub, C. van Loan, Matrix Computations. Johns Hopkins, 3rd edition, 1996. · Zbl 0865.65009
[11] Grisvard, P., Elliptic Problems in Nonsmooth Domains (1985), Pitman: Pitman London · Zbl 0695.35060
[12] Hardy, G.; Littlewood, J. E.; Pólya, G.; Cambridge Mathematical Library, Inequalities. (1991), Cambridge University Press: Cambridge University Press Cambridge
[13] Hu, N.; Guo, X.-Z.; Katz, I. N., Bounds for the eigenvalues and condition numbers in the \(p\)-version of the finite element method, Math. Comput., 67, 224, 1423-1450 (1998) · Zbl 0907.65112
[14] Hughes, T. J.R., The Finite Element Method (1987), Prentice-Hall: Prentice-Hall Englewood Cliffs, NY
[15] Maitre, J. F.; Pourquier, O., Condition number and diagonal preconditioning: comparison of the \(p\) version and the spectral element method, Numer. Math., 74, 69-84 (1996) · Zbl 0858.65114
[16] J.M. Melenk, K. Gerdes, C. Schwab, Fully discrete hp; J.M. Melenk, K. Gerdes, C. Schwab, Fully discrete hp · Zbl 0985.65141
[17] Olsen, E. T.; Douglas, J., Bounds on spectral condition numbers of matrices arising in the \(p\)-version of the finite element method, Numer. Math., 69, 333-352 (1995) · Zbl 0834.65034
[18] Raugel, G., Résolution numérique par une méthode d’élements finis du probleme de Dirichlet pour le Laplacien dans un polygone, C. R. Acad. Sci. Paris, 286, 791-794 (1978) · Zbl 0377.65058
[19] P. Schafheitlin, Über die Gaußsche und Besselsche Differentialgleichung und eine neue Integralform der letzteren, Journal für Mathematik 114 (1894) 31-44.; P. Schafheitlin, Über die Gaußsche und Besselsche Differentialgleichung und eine neue Integralform der letzteren, Journal für Mathematik 114 (1894) 31-44. · JFM 25.0839.01
[20] Schwab, C., \(p\)- and hp-Finite Element Methods (1998), Oxford University Press: Oxford University Press Oxford
[21] B. Sündermann, Lebesgue constants in Lagrangian interpolation at the Fekete points, Ergebnisberichte der Lehrstühle Mathematik III und VIII (Angewandte Mathematik) 44, Universität Dortmund, 1980.; B. Sündermann, Lebesgue constants in Lagrangian interpolation at the Fekete points, Ergebnisberichte der Lehrstühle Mathematik III und VIII (Angewandte Mathematik) 44, Universität Dortmund, 1980.
[22] Sündermann, B., Lebesgue constants in Lagrangian interpolation at the Fekete points, Mitt. Math. Ges. Hamb., 11, 204-211 (1983) · Zbl 0547.41002
[23] Szabó, B.; Babuška, I., Finite Element Analysis (1991), Wiley: Wiley New York
[24] B. Szegö, Orthogonal Polynomials, 4th Edition, American Mathematical Society, Providence, RI, 1975.; B. Szegö, Orthogonal Polynomials, 4th Edition, American Mathematical Society, Providence, RI, 1975.
[25] Szegö, G., Bestimmung derjenigen Abszissen eines Intervalles, für welche die Quadratsumme der Grundfunktionen der Lagrangeschen Interpolation im Intervalle ein möglichst kleines Maximum besitzt, Ann. Scuola Norm. Sup. Pisa, 1, 263-276 (1932) · Zbl 0004.24903
[26] G.N. Watson, A Treatise on the Theory of Bessel Functions, Cambridge University Press, second edition, 1995.; G.N. Watson, A Treatise on the Theory of Bessel Functions, Cambridge University Press, second edition, 1995. · Zbl 0849.33001
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.