Elliptic spline functions and the RayleighRitzGalerkin method
Author:
Martin H. Schultz
Journal:
Math. Comp. 24 (1970), 6580
MSC:
Primary 65.62; Secondary 41.00
MathSciNet review:
0264857
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Abstract: Error estimates for the RayleighRitzGalerkin method, using finitedimensional spline type spaces, for a class of nonlinear twopoint 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]
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Numer. Anal. 3 (1966), 173–182. MR 0217500
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Birkhoff, M.
H. Schultz, and R.
S. Varga, Piecewise Hermite interpolation in one and two variables
with applications to partial differential equations, Numer. Math.
11 (1968), 232–256. MR 0226817
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P.
G. Ciarlet, M.
H. Schultz, and R.
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boundary value problems. I. One dimensional problem, Numer. Math.
9 (1966/1967), 394–430. MR 0221761
(36 #4813)
 [4]
P.
G. Ciarlet, M.
H. Schultz, and R.
S. Varga, Numerical methods of highorder accuracy for nonlinear
boundary value problems. V. Monotone operator theory, Numer. Math.
13 (1969), 51–77. MR 0250496
(40 #3730)
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G.
H. Hardy, J.
E. Littlewood, and G.
Pólya, Inequalities, Cambridge, at the University
Press, 1952. 2d ed. MR 0046395
(13,727e)
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Philip
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Sons, Inc., New YorkLondonSydney, 1964. MR 0171038
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Bernie
L. Hulme, Interpolation by Ritz approximation, J. Math. Mech.
18 (1968/1969), 337–341. MR 0231537
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 [8]
G.
G. Lorentz, Approximation of functions, Holt, Rinehart and
Winston, New YorkChicago, Ill.Toronto, Ont., 1966. MR 0213785
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 [9]
F.
M. Perrin, H.
S. Price, and R.
S. Varga, On higherorder numerical methods for nonlinear twopoint
boundary value problems, Numer. Math. 13 (1969),
180–198. MR 0255069
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 [10]
Milton
E. Rose, Finite difference schemes for
differential equations, Math. Comp. 18 (1964), 179–195. MR 0183123
(32 #605), http://dx.doi.org/10.1090/S00255718196401831230
 [11]
Martin
H. Schultz, Error bounds for polynomial spline
interpolation, Math. Comp. 24 (1970), 507–515. MR 0275025
(43 #783), http://dx.doi.org/10.1090/S00255718197002750259
 [12]
M.
H. Schultz and R.
S. Varga, 𝐿splines, Numer. Math. 10
(1967), 345–369. MR 0225068
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 [1]
 J. H. Ahlberg & E. N. Nilson, "The approximation of linear functionals," SIAM J. Numer. Anal., v. 3, 1966, pp. 173182. 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. 232256. MR 37 #2404. MR 0226817 (37:2404)
 [3]
 P. G. Ciarlet, M. H. Schultz & R. S. Varga, "Numerical methods of highorder accuracy for nonlinear boundary value problems. I: One dimensional problem," Numer. Math., v. 9, 1967, pp. 394430. MR 36 #4813. MR 0221761 (36:4813)
 [4]
 P. G. Ciarlet, M. H. Schultz & R. S. Varga, "Numerical methods of highorder accuracy for nonlinear boundary value problems. V: Monotone operators," Numer. Math., v. 13, 1969, pp. 5177. 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. 337341. 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 higherorder numerical methods for nonlinear twopoint boundary value problems," Numer. Math., v. 13, 1968, pp. 180188. MR 0255069 (40:8276)
 [10]
 M. E. Rose, "Finite difference schemes for differential equations," Math. Comp., v. 18, 1964, pp. 179195. 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, "Lsplines," Numer. Math., v. 10, 1967, pp. 345369. MR 37 #665. MR 0225068 (37:665)
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00255718197002648579
PII:
S 00255718(1970)02648579
Keywords:
RayleighRitzGalerkin method,
variational method,
spline space,
twopoint boundary value problem,
spline interpolation
Article copyright:
© Copyright 1970
American Mathematical Society
