Arbitrarily Slow Convergence, Uniform Convergence and Superconvergence of Galerkin-like Methods

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Abstract
The first one is the class of methods which have a certain minimal order of convergence uniformly for all right-hand sides y. (This does not exclude that for some y there is a better order of convergence.) On the other hand, there are methods, which may have a good order of convergence for some y, but in general for each order of convergence there are y with a worse convergence rate. These methods converge arbitrarily slowly. This phenomenon is well known also in other fields of mathematics, for instance in approximation theory. Here the lethargy theorem of Bernstein states that, in each Banach space X, for each monotone decreasing sequence (ωn) and for each increasing sequence of finite-dimensional subspaces X. there is an x ε X such that dist(x,X)=ωn. (For converse results in this direction see Schock, 1971.) In this note I will show that the classical Galerkin method, and also, in some cases, the iterated Galerkin method of Sloan, converge arbitrarily slowly, on the other hand, that the Kantorovich method converges uniformly for all right-hand sides y. (Henceforth I shall use the term “converges uniformly” for brevity.)

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