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

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

MathSciNet review:
0433899

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Abstract | References | Similar Articles | Additional Information

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]**C. de BOOR, "On uniform approximation by splines,"*J. Approximation Theory*, v. 1, 1968, pp. 219-235. MR**39**#1866. MR**0240519 (39:1866)****[2]**C. de BOOR, "Good approximation by splines with variable knots,"*Conf. on the Numerical Solution of Differential Equations*(Scotland, 1973), Lecture Notes in Math., vol. 363, Springer-Verlag, Berlin and New York, 1974, pp. 12-20. MR**0431606 (55:4603)****[3]**C. de BOOR & B. SWARTZ, "Collocation at Gaussian points,"*SIAM J. Numer. Anal.*, v. 10, 1973, pp. 582-606. MR**51**#9528. MR**0373328 (51:9528)****[4]**J. DOUGLAS, JR., T. DUPONT & L. WAHLBIN, "Optimal error estimates for Galerkin approximations to solutions of two-point boundary value problems,"*Math. Comp.*, v. 29, 1975, pp. 475-483. MR**51**#7298. MR**0371077 (51:7298)****[5]**F. NATTERER, "Uber die punktweise Konvergenz Finiter Elemente,"*Numer. Math.*, v. 25, 1975, pp. 67-77. MR**0474884 (57:14514)****[6]**J. NITSCHE, " -convergence of finite element approximation. 2,"*Conf. on Finite Elements*(Rennes, France, 1975). (To appear.) MR**568857 (81e:65058)****[7]**R. SCOTT, "Optimal estimates for the finite element method on irregular meshes,"*Math. Comp.*, v. 30, 1976, pp. 681-697. MR**0436617 (55:9560)****[8]**M. F. WHEELER, "An optimal error estimate for Galerkin approximations to solutions of two-point boundary value problems,"*SIAM J. Numer. Anal.*, v. 10, 1973, pp. 914-917. MR**49**#8399. MR**0343659 (49:8399)**

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

Article copyright:
© Copyright 1977
American Mathematical Society