My last question was about the condition number of the stiffness matrix.  This question is about discretization error (which is probably more important to most people).  The mesh on the right side of the figure below (source) produced better discretization error than the perfect quality mesh on the left side even though the mesh on the right side has quads with angles close to 180 degrees. I assume the creator of the figure (professor Christopher Roy) used a finite volume or finite difference method since he is a fluid dynamics specialist.  Based on your personal experience using FEM, could FEM get a good discretization error using a mesh like the one on the right side of the figure?
Could the finite element method get a good discretization error using a mesh like this?
finite element;finite volume
I think, but I don't have the proof in front of me, that if you use a bilinear, Cartesian product element and the isoparametric map from the master element to the real element, then the Jacobian of the map will be zero at the flat corner on these elements. That might actually be OK if you never evaluate anything there. If you use Gaussian quadrature, then you may get away with only evaluating things in the interior at the element's Gauss points. If you use a Newton-Cotes rule that wants to do integration at the nodes of the element, then your ability to integrate at all is shot. So, the answer is probably it depends. In the Gaussian case, you may also lose some accuracy even if you don't evaluate anything.