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Mathematics of Computation
Mathematics of Computation
ISSN 1088-6842(online) ISSN 0025-5718(print)

 

Local piecewise polynomial projection methods for an O.D.E. which give high-order convergence at knots


Authors: Carl de Boor and Blair Swartz
Journal: Math. Comp. 36 (1981), 21-33
MSC: Primary 65L15
MathSciNet review: 595039
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Abstract: Local projection methods which yield $ {C^{(m - 1)}}$ piecewise polynomials of order $ m + k$ 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 $ \mathcal{O}(\vert\Delta {\vert^{2k}})$ superconvergence, namely (a) collocation at the k Gauss-Legendre points in each partition interval and (b) "essential least-squares" (i.e., local moment methods), is made by asking that this same order of superconvergence result when using collocation at $ k - r$ points per interval and simultaneous local orthogonality of the residual to polynomials of order r; the $ k - r$ points then necessarily form a subset of the k Gauss-Legendre points.


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DOI: http://dx.doi.org/10.1090/S0025-5718-1981-0595039-6
PII: S 0025-5718(1981)0595039-6
Keywords: Piecewise polynomial, projection methods, ordinary differential equations, boundary value problems, eigenvalue problems, superconvergence, ssuper projectors
Article copyright: © Copyright 1981 American Mathematical Society