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A fast algorithm for Gaussian type quadrature formulae with mixed boundary conditions and some lumped mass spectral approximations


Authors: Abdelkrim Ezzirani and Allal Guessab
Journal: Math. Comp. 68 (1999), 217-248
MSC (1991): Primary 65D30, 65D32, 65N35
DOI: https://doi.org/10.1090/S0025-5718-99-01001-7
MathSciNet review: 1604332
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Abstract: After studying Gaussian type quadrature formulae with mixed boundary conditions, we suggest a fast algorithm for computing their nodes and weights. It is shown that the latter are computed in the same manner as in the theory of the classical Gauss quadrature formulae. In fact, all nodes and weights are again computed as eigenvalues and eigenvectors of a real symmetric tridiagonal matrix. Hence, we can adapt existing procedures for generating such quadrature formulae. Comparative results with various methods now in use are given. In the second part of this paper, new algorithms for spectral approximations for second-order elliptic problems are derived. The key to the efficiency of our algorithms is to find an appropriate spectral approximation by using the most accurate quadrature formula, which takes the boundary conditions into account in such a way that the resulting discrete system has a diagonal mass matrix. Hence, our algorithms can be used to introduce explicit resolutions for the time-dependent problems. This is the so-called lumped mass method. The performance of the approach is illustrated with several numerical examples in one and two space dimensions.


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Additional Information

Abdelkrim Ezzirani
Affiliation: Laboratoire de Mathématiques Appliquées, UPRES A 5033, Associé au CNRS, Université de Pau, 64000, France

Allal Guessab
Affiliation: Laboratoire de Mathématiques Appliquées, UPRES A 5033, Associé au CNRS, Université de Pau, 64000, France
Email: guessab@univ-pau.fr

DOI: https://doi.org/10.1090/S0025-5718-99-01001-7
Keywords: Quadrature formulae, Gaussian quadrature formulae, spectral methods, lumped mass methods, quasi-orthogonal polynomials, algorithms
Received by editor(s): May 6, 1997
Additional Notes: The work of the second author was supported by the CNRS grant INTAS-94-4070.
Article copyright: © Copyright 1999 American Mathematical Society