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  • 1
    Electronic Resource
    Electronic Resource
    The convergence rates of expansions in Jacobi polynomials (1977)
    Springer
    Numerische Mathematik 27 (1977), S. 219-225 
    Details
    ISSN: 0945-3245
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary We consider the convergence of expansions of the form $$f\left( \chi \right) = \sum\limits_{n = 0}^\infty {b_n A_n P_n^{\left( {\alpha ,\beta } \right)} } \left( \chi \right)$$ whereA n P n (α,β) is then th normalised Jacobi polynomial. A modification of a technique described previously by Mead and Delves (1973) provides estimates of the convergence rate of the coefficientsb n which are more powerful than those so far available. The methods used are applicable to other orthogonal expansions, and the results obtained here are used in the companion paper (Delves and Bain (1976)) to discuss the optimum choice of weight functions in a class of variational methods.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01396642
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    Paper (German National Licenses)
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  • 2
    Electronic Resource
    Electronic Resource
    On the optimum choice of weight functions in a class of variational calculations (1977)
    Springer
    Numerische Mathematik 27 (1977), S. 209-218 
    Details
    ISSN: 0945-3245
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary We consider the variational solution of a particular class of second order differential equations and show that expansions in terms of Chebychev and a range of ultraspherical polynomials lead to operator matrices that are asymptotically diagonal, and that hence their convergence properties can be completely characterised using a previously developed analysis. For a given class of weight functions bounds are given on the convergence of the coefficients and of the weighted mean square error, in terms of the analyticity properties of the coefficients in the differential equation. These bounds are used to discuss the optimum choice of weight function for such a calculation.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01396641
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    Paper (German National Licenses)
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  • 3
    Electronic Resource
    Electronic Resource
    On the least squares procedure for atomic calculations (1969)
    New York, NY : Wiley-Blackwell
    International Journal of Quantum Chemistry 3 (1969), S. 169-184 
    Details
    ISSN: 0020-7608
    Keywords: Computational Chemistry and Molecular Modeling ; Atomic, Molecular and Optical Physics
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Chemistry and Pharmacology
    Notes: We discuss the properties of one version of the least squares (LS) method for the solution of the Schrödinger equation. These properties are exemplified by a number of calculations on the n1S and n3S states of helium, up to principal quantum number three, which are very much more accurate than previous LS calculations on helium. Particular attention is paid to the convergence properties of the LS procedure and we compare it with the simpler Rayleigh-Ritz (RR) procedure in the case when the RR matrix elements are evaluated numerically over the same quadrature mesh as used in the LS procedure. We conclude that although the LS procedure is capable of high accuracy it has no advantages which would justify its sole use in place of the RR procedure. However, it does have some advantages when used in conjunction with RR, in that it gives an estimate of the numerical accuracy of the RR energies.
    Additional Material: 6 Ill.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1002/qua.560030203
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    Paper (German National Licenses)
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  • 4
    Electronic Resource
    Electronic Resource
    The global element method for stationary advective problems (1980)
    Chichester [u.a.] : Wiley-Blackwell
    International Journal for Numerical Methods in Engineering 15 (1980), S. 167-175 
    Details
    ISSN: 0029-5981
    Keywords: Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics , Technology
    Notes: We extend the variation principle used in the global element method for self-adjoint elliptic problems1, to problems containing advective terms. Used with the global element formalism, the extended principle yields a uniform framework for treating advective or boundary layer problems, with the attractive feature that the implicit treatment of interface conditions between elements yields an effective decoupling between ‘boundary layer’ and ‘interior’ parts of the solution. As a numerical example, we solve the one-dimensional model problem of Christie and Mitchell3, obtaining high accuracy for Peclet numbers up to 106 with no sign of instability. These results suggest that, given a suitable choice of global elements, the decoupling is very effective in damping the oscillations found in standard finite difference or finite element treatment.
    Additional Material: 2 Tab.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1002/nme.1620150202
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  • 5
    Electronic Resource
    Electronic Resource
    Performance of GEM2 on the ELLPACK problem population (1986)
    Chichester [u.a.] : Wiley-Blackwell
    International Journal for Numerical Methods in Engineering 23 (1986), S. 229-238 
    Details
    ISSN: 0029-5981
    Keywords: Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics , Technology
    Notes: The Pilot global element program GEM2 MK1.0 has been used to solve the two-dimensional problems in the ELLPACK population of partial differential equation test problems. The performance achieved is summarized in this paper; special attention is paid to those problems which are singular or near singular, and for which comparisons have been published by Houstis and Rice.12
    Additional Material: 5 Ill.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1002/nme.1620230207
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  • 6
    Electronic Resource
    Electronic Resource
    Discontinuities across element interfaces in the global element method (1986)
    New York, NY [u.a.] : Wiley-Blackwell
    Numerical Methods for Partial Differential Equations 2 (1986), S. 259-272 
    Details
    ISSN: 0749-159X
    Keywords: Mathematics and Statistics ; Numerical Methods
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Notes: In this article we extend the global element method formalism (Delves and Hall, 1979) for two-dimensional partial differential equations to admit the possibility of solution discontinuities across element interfaces. This extension has particular relevance to problems involving sinks and sources, such as occur in oil well or groundwater management.
    Additional Material: 3 Ill.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1002/num.1690020403
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    Paper (German National Licenses)
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