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Mathematics of Computation

ISSN 1088-6842(online) ISSN 0025-5718(print)



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

$\displaystyle {\left\Vert {y - {y^\pi }} \right\Vert _{{L_\infty }(a,b)}} \leqs... ...its_i h_i^{r + 1}{\left\Vert {{y^{(r + 1)}}} \right\Vert _{{L_\infty }({I_i})}}$

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