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 \[ {\left \| {y - {y^\pi }} \right \|_{{L_\infty }(a,b)}} \leqslant C\max \limits _i h_i^{r + 1}{\left \| {{y^{(r + 1)}}} \right \|_{{L_\infty }({I_i})}}\] to hold are given. Here, ${y^\pi }$ is the Galerkin approximation to the solution *y* of a boundary value problem for an ordinary differential equation, the trial functions being polynomials of degree $\leqslant r$ on the subintervals ${I_i} = [{x_i},{x_{i + 1}}]$ of length ${h_i}$. The sequence of subdivisions $\pi :{x_0} < {x_1} < \cdots < {x_n}$ need not be quasi-uniform.

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Article copyright:
© Copyright 1977
American Mathematical Society