Let $A_h \in \mathbb{R}^{n \times n}$ be the matrix corresponding to a finite element discretization of some nonselfadjoint, bounded and indefinite bilinear form corresponding to a second order operator satisfying a Garding inequality.

Let $A_H \in \mathbb{R}^{m \times m}$ be a matrix corresponding to the same finite element discretization on a coarser grid (grids are nested) and let $R_H$ be the $L_2$-projection from $h$ to $H$.

What $L_2$ bounds can I get for the numerical range of $R_H^T A_H^{-1} R_H A_h$ as a function of $h$ and $H$?

I am using a nonoverlapping symmetric interior penalty discontinuous Galerkin discretization, if it is relevant.