A GENERAL FRAMEWORK FOR THE RIGOROUS COMPUTATION OF INVARIANT DENSITIES AND THE COARSE-FINE STRATEGY

The paper’s Theorem 3.4 states the same error bound as the model’s result and is, in substance, correct; however, the proof as written contains a gap. In the proof, the paper sets v = u − u_h ∈ U_h^0 and then applies the telescoping estimate ||v|| ≤ ∑_k ||Q_h^k(Q_h v − v)|| ≤ (∑_k C_k) ||Q_h v − v||, together with lim_m ||Q_h^m v|| → 0, but this requires v ∈ U_h^0, which need not hold because u need not lie in U_h (see the definition of U_h^0 and the statement of Theorem 3.4) . The model fixes this by constructing g = u_h − P_h u + c_h·1 ∈ U_h^0 and applying the geometric-series identity on U_h^0, which legitimately uses the bounds ||Q_h^k|_{U_h^0}|| ≤ C_k. The remaining ingredients (e.g., bounding (I − Q_h)P_h u and the need for a bound on ||L||) are explicitly part of the paper’s framework and constants list . Hence the theorem’s conclusion is correct, but the paper’s proof needs a small repair, which the model supplies.