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

Natterer, Frank

doi:10.1090/S0025-5718-1977-0433899-9

Uniform convergence of Galerkin’s method for splines on highly nonuniform meshes
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Math. Comp. 31 (1977), 457-468 Request permission

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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