Asymptotic rates of convergence - for quadrature, ODEs and PDEs

The asymptotic rate of convergence of the trapezium rule is

defined, for smooth functions, by the Euler-Maclaurin expansion.

The extension to other methods, such as Gauss rules, is straightforward;

this talk begins with some special cases, such as Periodic functions, and

functions with various singularities.

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Convergence rates for ODEs (Initial and Boundary value problems)

and for PDEs are available, but not so well known. Extension to singular

problems seems to require methods specific to each situation. Some of

the results are unexpected - to me, anyway.