  The paper starts out with a computational technique that allows to compare
all linear methods for PDE solving that use the same input data. This is done
by writing them as linear recovery formulas for solution values as linear
combinations of the input data. Calculating the norm of these reproduction
formulas on a fixed Sobolev space will then serve as a quality criterion that
allows a fair comparison of all linear methods with the same inputs, including
finite-element, finite-difference and meshless local Petrov-Galerkin
techniques. A number of illustrative examples will be provided. As a byproduct,
it turns out that a unique error--optimal method exists. It necessarily
outperforms any other competing technique using the same data, e.g. those just
mentioned, and it is necessarily meshless, if solutions are written "entirely
in terms of nodes" (Belytschko et. al. 1996). On closer inspection, it turns
out that it coincides with {\em symmetric meshless collocation} carried out
with the kernel of the Hilbert space used for error evaluation, e.g. with the
kernel of the Sobolev space used. This technique is around since at least 1998,
but its optimality properties went unnoticed, so far.
