Claudio Pedrazzi

my two cents: in my opinion even looking at the stresses in the elements you can't avoid stress concentrations. If you shrink the mesh even the stresses in the elements (which, by the way, are the ones that are calculated first, at the Gauss points, from the nodal deformations, which are the real "solution" of the FEM system of equation: then they are extrapolated to the nodes), they too will grow indefinitely.

If the real stress state at that location is important to you (i.e., you can't do your maximization excluding that location), I personally see no alternative but to use a material model that is a little more realistic than linear: at a minimum, bilinear. That way the peaks will eliminate themselves, as they do in physical reality, generating microzones of plasticity.