Clearly, best
is a subjective term. If you are
looking for the wave function within a definite set that has the most
accurate expectation value of energy, then minimizing the expectation
value of energy will do it. This function will also approximate the
true eigenfunction shape the best, in some technical sense
{A.7}. (There are many ways the best approximation of
a function can be defined; you can demand that the maximum error is as
small as possible, or that the average magnitude of the error is as
small as possible, or that a root-mean-square error is, etcetera. In
each case, the best
answer will be different, though
there may not be much of a practical difference.)
But given a set of approximate wave functions like those used in finite element methods, it may well be possible to get much better results using additional mathematical techniques like Richardson extrapolation. In effect you are then deducing what happens for wave functions that are beyond the approximate ones you are using.