  We propose a family of gradient reconstruction schemes based on the solution
of over-determined systems by orthogonal or oblique projections. In the case of
orthogonal projections, we retrieve familiar weighted least-squares gradients,
but we also propose new direction-weighted variants. On the other hand, using
oblique projections that employ cell face normal vectors we derive variations
of consistent Green-Gauss gradients, which we call Taylor-Gauss gradients. The
gradients are tested and compared on a variety of grids such as structured,
locally refined, randomly perturbed, unstructured, and with high aspect ratio.
The tests include quadrilateral and triangular grids, and employ both compact
and extended stencils, and observations are made about the best choice of
gradient and weighting scheme for each case. On high aspect ratio grids, it is
found that most gradients can exhibit a kind of numerical instability that may
be so severe as to make the gradient unusable. A theoretical analysis of the
instability reveals that it is triggered by roundoff errors in the calculation
of the cell centroids, but ultimately is due to truncation errors of the
gradient reconstruction scheme, rather than roundoff errors. Based on this
analysis, we provide guidelines on the range of weights that can be used safely
with least squares methods to avoid this instability.
