Are bilinear quadrilaterals better than linear triangles?

This paper compares the theoretical effectiveness of bilinear approximation over quadrilaterals with linear approximation over triangles. Anisotropic mesh transformation is used to generate asymptotically optimally efficient meshes for piecewise linear interpolation over triangles and bilinear interpolation over quadrilaterals. For approximating a convex function, although bilinear quadrilaterals are more efficient, linear triangles are more accurate and may be preferred in finite element computations; whereas for saddle-shaped functions, quadrilaterals may over a higher-order approximation on a well-designed mesh. A surprising finding is different grid orientations may yield an order of magnitude improvement in approximation accuracy.