Tight Tractability Results for a Model Second-Order Neumann Problem

We study the worst case complexity and the tractability of the model second-order problem $$-\Delta u+u=f$$-Δu+u=f on $$I^d=(0,1)^d $$Id=(0,1)d with homogeneous Neumann boundary conditions. As is often the case, we study the variational formulation of this problem. Previous work on the tractability of problems such as this relied on the fact that such problems were reducible to the $$L_2(I^d)$$L2(Id)-approximation problem, which allowed us to find necessary and sufficient conditions for the problem to have a given degree of tractability. However, such an approach can only yield sufficient conditions, and not necessary conditions, for the model second-order Neumann problem to have a given degree of tractability. In this paper, we remedy this gap and find necessary and sufficient conditions for this Neumann problem to exhibit a given degree of tractability.