Local piecewise polynomial projection methods for an O.D.E. which give highorder convergence at knots
Authors:
Carl de Boor and Blair Swartz
Journal:
Math. Comp. 36 (1981), 2133
MSC:
Primary 65L15
MathSciNet review:
595039
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Abstract: Local projection methods which yield piecewise polynomials of order as approximate solutions of a boundary value problem for an mth order ordinary differential equation are determined by the k linear functional at which the residual error in each partition interval is required to vanish on. We develop a condition on these k functionals which implies breakpoint superconvergence (of derivatives of order less than m) for the approximating piecewise polynomials. The same order of superconvergence is associated with eigenvalue problems. A discrete connection between two particular projectors yielding superconvergence, namely (a) collocation at the k GaussLegendre points in each partition interval and (b) "essential leastsquares" (i.e., local moment methods), is made by asking that this same order of superconvergence result when using collocation at points per interval and simultaneous local orthogonality of the residual to polynomials of order r; the points then necessarily form a subset of the k GaussLegendre points.
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 [4]
 S. A. Pruess, "Solving linear boundary value problems by approximating the coefficients," Math. Comp., v. 27, 1973, pp. 551561. MR 0371100 (51:7321)
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 S. A. Pruess, "High order approximation to SturmLiouville eigenvalues," Numer. Math., v. 24, 1975, pp. 241247. MR 0378431 (51:14599)
 [6]
 K. A. Wittenbrink, "High order projection methods of moment and collocation type for nonlinear boundary value problems," Computing, v. 11, 1973, pp. 255274. MR 0400724 (53:4554)
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00255718198105950396
PII:
S 00255718(1981)05950396
Keywords:
Piecewise polynomial,
projection methods,
ordinary differential equations,
boundary value problems,
eigenvalue problems,
superconvergence,
ssuper projectors
Article copyright:
© Copyright 1981
American Mathematical Society
