A fast algorithm for Gaussian type quadrature formulae with mixed boundary conditions and some lumped mass spectral approximations. (English) Zbl 0916.65020
Authors’ 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, a 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.
In the second part of this paper, a 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.
Reviewer: R.Scherer (Karlsruhe)
MSC:
| 65D32 | Numerical quadrature and cubature formulas |
| 65N35 | Spectral, collocation and related methods for boundary value problems involving PDEs |
| 41A55 | Approximate quadratures |
| 35J25 | Boundary value problems for second-order elliptic equations |
Keywords:
spectral methods; lumped mass methods; quasi-orthogonal polynomials; Gaussian type quadrature formulae; fast algorithm; nodes; weights; eigenvalues; eigenvectors; tridiagonal matrix; second-order elliptic problems; performance; numerical examplesSoftware:
ORTHPOLReferences:
| [1] | Richard Askey, Positive quadrature methods and positive polynomial sums, Approximation theory, V (College Station, Tex., 1986) Academic Press, Boston, MA, 1986, pp. 1 – 29. · Zbl 0613.41027 |
| [2] | P. Berckmann, Orthogonal polynomials for engineers and physicists, The Golem Press, Boulder, Colorado, 1973. |
| [3] | Christine Bernardi and Yvon Maday, Some spectral approximations of one-dimensional fourth-order problems, Progress in approximation theory, Academic Press, Boston, MA, 1991, pp. 43 – 116. · Zbl 0701.41009 |
| [4] | Christine Bernardi and Yvon Maday, Approximations spectrales de problèmes aux limites elliptiques, Mathématiques & Applications (Berlin) [Mathematics & Applications], vol. 10, Springer-Verlag, Paris, 1992 (French, with French summary). · Zbl 0773.47032 |
| [5] | Borislav D. Bojanov, Dietrich Braess, and Nira Dyn, Generalized Gaussian quadrature formulas, J. Approx. Theory 48 (1986), no. 4, 335 – 353. · Zbl 0607.41009 · doi:10.1016/0021-9045(86)90008-0 |
| [6] | Borislav Bojanov and Geno Nikolov, Comparison of Birkhoff type quadrature formulae, Math. Comp. 54 (1990), no. 190, 627 – 648. · Zbl 0698.65015 |
| [7] | B. Bojanov, G. Grozev and A. A. Zhensykbaev, Generalized Gaussian quadrature formulas for weak Chebychev systems, in Optimal Recovery of Functions, B. Bojanov and H. Wozniakowski , Nova Sciences, New York, 1992, pp. 115-140. · Zbl 0779.41010 |
| [8] | B. Bojanov and A. Guessab, Gaussian quadrature formula of Birkhoff’s type, to appear. · Zbl 0918.41026 |
| [9] | Claudio Canuto, M. Yousuff Hussaini, Alfio Quarteroni, and Thomas A. Zang, Spectral methods in fluid dynamics, Springer Series in Computational Physics, Springer-Verlag, New York, 1988. · Zbl 0658.76001 |
| [10] | Philippe G. Ciarlet, The finite element method for elliptic problems, North-Holland Publishing Co., Amsterdam-New York-Oxford, 1978. Studies in Mathematics and its Applications, Vol. 4. · Zbl 0383.65058 |
| [11] | Philip J. Davis and Philip Rabinowitz, Methods of numerical integration, 2nd ed., Computer Science and Applied Mathematics, Academic Press, Inc., Orlando, FL, 1984. · Zbl 0537.65020 |
| [12] | E. A. Van Doorn, Representation and bounds for zeros of orthogonal polynomials and eigenvalues of sign-symmetric tri-diagonal matrices, J. Approx. Theory, v. 51, 1987, pp. 254-266. · Zbl 0636.42023 |
| [13] | A. Ezzirani, Construction de formules de quadrature pour des systèmes de Chebychev avec applications aux méthodes spectrales, Thèse de l’Université de Pau, France, 1996. |
| [14] | A. Ezzirani and A. Guessab, A fast algorithm for Gaussian type quadrature formulae with mixed boundary conditions, and some lumped mass spectral approximations, U.A. CNRS 1204. 6(1998). · Zbl 0916.65020 |
| [15] | Daniele Funaro, Polynomial approximation of differential equations, Lecture Notes in Physics. New Series m: Monographs, vol. 8, Springer-Verlag, Berlin, 1992. · Zbl 0774.41010 |
| [16] | Walter Gautschi, A survey of Gauss-Christoffel quadrature formulae, E. B. Christoffel (Aachen/Monschau, 1979) Birkhäuser, Basel-Boston, Mass., 1981, pp. 72 – 147. |
| [17] | Walter Gautschi and Shikang Li, Gauss-Radau and Gauss-Lobatto quadratures with double end points, J. Comput. Appl. Math. 34 (1991), no. 3, 343 – 360. · Zbl 0727.65012 · doi:10.1016/0377-0427(91)90094-Z |
| [18] | W. Gautschi, Algorithm 726: ORTHPOL- A package of routines for generating orthogonal polynomials and Gauss-type quadrature rules, ACM Trans. Math. Software 20(1994), 21-62. · Zbl 0888.65013 |
| [19] | Gene H. Golub and John H. Welsch, Calculation of Gauss quadrature rules, Math. Comp. 23 (1969), 221-230; addendum, ibid. 23 (1969), no. 106, loose microfiche suppl, A1 – A10. · Zbl 0179.21901 |
| [20] | Gene H. Golub, Some modified matrix eigenvalue problems, SIAM Rev. 15 (1973), 318 – 334. · Zbl 0254.65027 · doi:10.1137/1015032 |
| [21] | G. H. Golub and J. Kautský, Calculation of Gauss quadratures with multiple free and fixed knots, Numer. Math. 41 (1983), no. 2, 147 – 163. · Zbl 0525.65010 · doi:10.1007/BF01390210 |
| [22] | A. Guessab and Q. I. Rahman, Quadrature formulae and polynomial inequalities, J. Approx. Theory, v. 90, 1997, pp. 255-282. CMP 97:16 · Zbl 0879.41014 |
| [23] | A. Guessab and G. V. Milovanović, An algorithm for Gauss-Birkhoff type quadrature formulae, to appear. |
| [24] | Thomas J. R. Hughes, The finite element method, Prentice Hall, Inc., Englewood Cliffs, NJ, 1987. Linear static and dynamic finite element analysis; With the collaboration of Robert M. Ferencz and Arthur M. Raefsky. · Zbl 0634.73056 |
| [25] | Kurt Jetter, A new class of Gaussian quadrature formulas based on Birkhoff type data, SIAM J. Numer. Anal. 19 (1982), no. 5, 1081 – 1089. · Zbl 0496.41018 · doi:10.1137/0719078 |
| [26] | Samuel Karlin and William J. Studden, Tchebycheff systems: With applications in analysis and statistics, Pure and Applied Mathematics, Vol. XV, Interscience Publishers John Wiley & Sons, New York-London-Sydney, 1966. · Zbl 0153.38902 |
| [27] | M. G. Kreĭn, The ideas of P. L. Čebyšev and A. A. Markov in the theory of limiting values of integrals and their further development, Amer. Math. Soc. Transl. (2) 12 (1959), 1 – 121. M. G. Kreĭn and P. G. Rehtman, Development in a new direction of the Čebyšev-Markov theory of limiting values of integrals, Amer. Math. Soc. Transl. (2) 12 (1959), 123 – 135. |
| [28] | George G. Lorentz, Kurt Jetter, and Sherman D. Riemenschneider, Birkhoff interpolation, Encyclopedia of Mathematics and its Applications, vol. 19, Addison-Wesley Publishing Co., Reading, Mass., 1983. · Zbl 0522.41001 |
| [29] | Avraham A. Melkman, Interpolation by splines satisfying mixed boundary conditions, Israel J. Math. 19 (1974), 369 – 381. · Zbl 0303.41007 · doi:10.1007/BF02757500 |
| [30] | C. A. Micchelli and Allan Pinkus, Moment theory for weak Chebyshev systems with applications to monosplines, quadrature formulae and best one-sided \?\textonesuperior -approximation by spline functions with fixed knots, SIAM J. Math. Anal. 8 (1977), no. 2, 206 – 230. · Zbl 0353.41003 · doi:10.1137/0508015 |
| [31] | C. A. Micchelli and T. J. Rivlin, Numerical integration rules near Gaussian quadrature, Israel J. Math. 16 (1973), 287 – 299. · Zbl 0287.65014 · doi:10.1007/BF02756708 |
| [32] | C. A. Micchelli and T. J. Rivlin, Quadrature formulae and Hermite-Birkhoff interpolation, Advances in Math. 11 (1973), 93 – 112. · Zbl 0259.41016 · doi:10.1016/0001-8708(73)90004-2 |
| [33] | Gradimir V. Milovanović, Construction of \?-orthogonal polynomials and Turán quadrature formulae, Numerical methods and approximation theory, III (Niš, 1987) Univ. Niš, Niš, 1988, pp. 311 – 328. · Zbl 0643.65011 |
| [34] | Beresford N. Parlett, The symmetric eigenvalue problem, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1980. Prentice-Hall Series in Computational Mathematics. · Zbl 0431.65017 |
| [35] | Franz Peherstorfer, Characterization of positive quadrature formulas, SIAM J. Math. Anal. 12 (1981), no. 6, 935 – 942. · Zbl 0481.41025 · doi:10.1137/0512079 |
| [36] | Franz Peherstorfer, Characterization of quadrature formula. II, SIAM J. Math. Anal. 15 (1984), no. 5, 1021 – 1030. · Zbl 0596.41044 · doi:10.1137/0515079 |
| [37] | Alfio Quarteroni and Alberto Valli, Numerical approximation of partial differential equations, Springer Series in Computational Mathematics, vol. 23, Springer-Verlag, Berlin, 1994. · Zbl 0803.65088 |
| [38] | Theodore J. Rivlin, Chebyshev polynomials, 2nd ed., Pure and Applied Mathematics (New York), John Wiley & Sons, Inc., New York, 1990. From approximation theory to algebra and number theory. · Zbl 0734.41029 |
| [39] | H. J. Schmid, A note on positive quadrature rules, Rocky Mountain J. Math. 19 (1989), no. 1, 395 – 404. Constructive Function Theory — 86 Conference (Edmonton, AB, 1986). · Zbl 0691.41032 · doi:10.1216/RMJ-1989-19-1-395 |
| [40] | D. D. Stancu, Sur quelques formules générales de quadrature du type Gauss-Christoffel, Mathematica (Cluj) 1 (24) (1959), no. 1, 167 – 182 (French). · Zbl 0094.11503 |
| [41] | Gábor Szegő, Orthogonal polynomials, 4th ed., American Mathematical Society, Providence, R.I., 1975. American Mathematical Society, Colloquium Publications, Vol. XXIII. · JFM 65.0278.03 |
| [42] | David S. Watkins, Some perspectives on the eigenvalue problem, SIAM Rev. 35 (1993), no. 3, 430 – 471. · Zbl 0786.65032 · doi:10.1137/1035090 |
| [43] | Yuan Xu, Quasi-orthogonal polynomials, quadrature, and interpolation, J. Math. Anal. Appl. 182 (1994), no. 3, 779 – 799. · Zbl 0802.42020 · doi:10.1006/jmaa.1994.1121 |
| [44] | Yuan Xu, A characterization of positive quadrature formulae, Math. Comp. 62 (1994), no. 206, 703 – 718. · Zbl 0799.65020 |
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.
