  We study gauge fixing via the standard local extremization algorithm for
2-dimensional $U(1)$. On a lattice with spherical topology $S^2$ where all
copies are lattice artifacts, we find that the number of these 'Gribov' copies
diverges in the continuum limit. On a torus, we show that lattice artifacts can
lead to the wrong evaluation of the gauge-invariant correlation length, when
measured via a gauge-fixed procedure; this bias does not disappear in the
continuum limit. We then present a new global approach, based on Hodge
decomposition of the gauge field, which produces a unique smooth field in
Landau gauge, and is economically powered by the FFT. We also discuss the use
of this method for examining topological objects, and its extensions to
non-abelian gauge fields.
