  We present quantitative means for assessing the numerical accuracy of static
magnetic field calculations in finite-element models. Our calculations use the
three-dimensional Opera simulation software suite of Dassault Systemes. Our
need to assess the effects of fringe fields requires such a 3D algorithm. While
we do discuss and compare our approach to a method of accuracy estimation used
in the Opera post-processor, our methods are generally applicable to any model
using relaxation techniques in finite-element systems. For purposes of
illustration, we present modeling and analysis of two types of quadrupole
electromagnets presently in operation in the south arc of the Cornell Electron
Storage Ring (CESR). Calculations of field multipole expansion coefficients and
numerical deviations from Maxwell's equations in source-free regions are
discussed, with emphasis on the dependence of their accuracy on changes to the
finite-element model. Successive refinement steps in the finite-element model
for the non-extraction type of CESR south arc quadrupole achieve a reduction in
the RMS value of the longitudinal component of the curl vector on the magnet
axis by a factor of nearly 70 from $4.38 \times 10^{-2}$ T/m to $6.36 \times
10^{-4}$ T/m, which is 0.0021% of the field gradient. An accuracy of 2.4% is
achieved for a dodecapole coefficient of $2.4 \times 10^{-4}$ of that of the
quadrupole in the longitudinal integral of the vertical field gradient at a
radius of 1 cm.
