  In our previous works, we proved that the inverse of the stiffness matrix of
an $h$-version finite element method (FEM) applied to scalar second order
elliptic boundary value problems can be approximated at an exponential rate in
the block rank by $\mathcal{H}$-matrices. Here, we improve on this result in
multiple ways: (1) The class of meshes is significantly enlarged and includes
certain exponentially graded meshes. (2) The dependence on the polynomial
degree $p$ of the discrete ansatz space is made explicit in our analysis. (3)
The bound for the approximation error is sharpened, and (4) the proof is
simplified.
