Elliptic spline functions and the Rayleigh-Ritz-Galerkin method

Author:
Martin H. Schultz

Journal:
Math. Comp. **24** (1970), 65-80

MSC:
Primary 65.62; Secondary 41.00

DOI:
https://doi.org/10.1090/S0025-5718-1970-0264857-9

MathSciNet review:
0264857

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Abstract | References | Similar Articles | Additional Information

Abstract: Error estimates for the Rayleigh-Ritz-Galerkin method, using finite-dimensional spline type spaces, for a class of nonlinear two-point boundary value problems are discussed. The results of this paper extend and improve recent corresponding results of B. L. Hulme, F. M. Perrin, H. S. Price, and R. S. Varga.

**[1]**J. H. Ahlberg & E. N. Nilson, "The approximation of linear functionals,"*SIAM J. Numer. Anal.*, v. 3, 1966, pp. 173-182. MR**36**#589. MR**0217500 (36:589)****[2]**G. Birkhoff, M. H. Schultz & R. S. Varga, "Piecewise Hermite interpolation in one and two variables with applications to partial differential equations,"*Numer. Math.*, v. 11, 1968, pp. 232-256. MR**37**#2404. MR**0226817 (37:2404)****[3]**P. G. Ciarlet, M. H. Schultz & R. S. Varga, "Numerical methods of high-order accuracy for nonlinear boundary value problems. I: One dimensional problem,"*Numer. Math.*, v. 9, 1967, pp. 394-430. MR**36**#4813. MR**0221761 (36:4813)****[4]**P. G. Ciarlet, M. H. Schultz & R. S. Varga, "Numerical methods of high-order accuracy for nonlinear boundary value problems. V: Monotone operators,"*Numer. Math.*, v. 13, 1969, pp. 51-77. MR**0250496 (40:3730)****[5]**G. H. Hardy, J. E. Littlewood & G. Pólya,*Inequalities*, 2nd ed., Cambridge Univ. Press, New York, 1952. MR**13**, 727. MR**0046395 (13:727e)****[6]**P. Hartman,*Ordinary Differential Equations*, Wiley, New York, 1964. MR**30**#1270. MR**0171038 (30:1270)****[7]**B. L. Hulme, "Interpolation by Ritz approximation,"*J. Math. Mech.*, v. 18, 1968, pp. 337-341. MR**37**#7090. MR**0231537 (37:7090)****[8]**G. G. Lorentz,*Approximation of Functions*, Holt, Rinehart, and Winston, New York, 1966. MR**35**#4642. MR**0213785 (35:4642)****[9]**F. M. Perrin, H. S. Price & R. S. Varga, "On higher-order numerical methods for nonlinear two-point boundary value problems,"*Numer. Math.*, v. 13, 1968, pp. 180-188. MR**0255069 (40:8276)****[10]**M. E. Rose, "Finite difference schemes for differential equations,"*Math. Comp.*, v. 18, 1964, pp. 179-195. MR**32**#605. MR**0183123 (32:605)****[11]**M. H. Schultz, "Error bounds for polynomial spline interpolation." (To appear.) MR**0275025 (43:783)****[12]**M. H. Schultz & R. S. Varga, "*L*-splines,"*Numer. Math.*, v. 10, 1967, pp. 345-369. MR**37**#665. MR**0225068 (37:665)**

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Additional Information

DOI:
https://doi.org/10.1090/S0025-5718-1970-0264857-9

Keywords:
Rayleigh-Ritz-Galerkin method,
variational method,
spline space,
two-point boundary value problem,
spline interpolation

Article copyright:
© Copyright 1970
American Mathematical Society