Uniform convergence of Galerkin's method for splines on highly nonuniform meshes

Author:
Frank Natterer

Journal:
Math. Comp. **31** (1977), 457-468

MSC:
Primary 65L10

MathSciNet review:
0433899

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Abstract: Different sets of conditions for an estimate of the form

*y*of a boundary value problem for an ordinary differential equation, the trial functions being polynomials of degree on the subintervals of length .

The sequence of subdivisions need not be quasi-uniform.

**[1]**Carl de Boor,*On uniform approximation by splines*, J. Approximation Theory**1**(1968), 219–235. MR**0240519****[2]**Carl de Boor,*Good approximation by splines with variable knots. II*, Conference on the Numerical Solution of Differential Equations (Univ. Dundee, Dundee, 1973) Springer, Berlin, 1974, pp. 12–20. Lecture Notes in Math., Vol. 363. MR**0431606****[3]**Carl de Boor and Blâir Swartz,*Collocation at Gaussian points*, SIAM J. Numer. Anal.**10**(1973), 582–606. MR**0373328****[4]**Jim Douglas Jr., Todd Dupont, and Lars Wahlbin,*Optimal 𝐿_{∞} error estimates for Galerkin approximations to solutions of two-point boundary value problems*, Math. Comp.**29**(1975), 475–483. MR**0371077**, 10.1090/S0025-5718-1975-0371077-0**[5]**Frank Natterer,*Über die punktweise Konvergenz finiter Elemente*, Numer. Math.**25**(1975/76), no. 1, 67–77 (German, with English summary). MR**0474884****[6]**J. A. Nitsche,*𝐿_{∞}-convergence of finite element approximation*, Journées “Éléments Finis” (Rennes, 1975) Univ. Rennes, Rennes, 1975, pp. 18. MR**568857****[7]**Ridgway Scott,*Optimal 𝐿^{∞} estimates for the finite element method on irregular meshes*, Math. Comp.**30**(1976), no. 136, 681–697. MR**0436617**, 10.1090/S0025-5718-1976-0436617-2**[8]**Mary Fanett Wheeler,*An optimal 𝐿_{∞} error estimate for Galerkin approximations to solutions of two-point boundary value problems*, SIAM J. Numer. Anal.**10**(1973), 914–917. MR**0343659**

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DOI:
https://doi.org/10.1090/S0025-5718-1977-0433899-9

Article copyright:
© Copyright 1977
American Mathematical Society