On the convergence rates of variational methods. I. Asymptotically diagonal systems

Abstract:

We consider the problem of estimating the convergence rate of a variational solution to an inhomogeneous equation. This problem is not soluble in general without imposing conditions on both the class of expansion functions and the class of problems considered; we introduce the concept of “asymptotically diagonal systems,” which is particularly appropriate for classical variational expansions as applied to elliptic partial differential equations. For such systems, we obtain a number of a priori estimates of the asymptotic convergence rate which are easy to compute, and which are likely to be realistic in practice. In the simplest cases these estimates reduce the problem of variational convergence to the simpler problem of Fourier series convergence, which is considered in a companion paper. We also produce estimates for the convergence rate of the individual expansion coefficients $a_i^{(n)}$, thus categorising the convergence completely.