Complexity for a class of elliptic ordinary integro-differential equations

Consider the variational form of the ordinary integro-differential equation (OIDE)$-u^{″}+u+\underset{0}{\overset{1}{\int }}q(\cdot ,y)u\left(y\right)\text{dy}=f$ on the unit interval I, subject to homogeneous Neumann boundary conditions. Here, f and q respectively belong to the unit ball of $H^r\left(I\right)$ and the ball of radius $M_1$ of $H^s\left(I^2\right)$, where $M_1\in [0,1)$. For $\epsilon >0$, we want to compute ε-approximations for this problem, measuring error in the $H^1\left(I\right)$ sense in the worst case setting. Assuming that standard information is admissible, we find that the nth minimal error is $\Theta \left(n^{-\min \{r,s/2\}}\right)$, so that the information ε-complexity is $\Theta \left({\epsilon }^{-1/\min \{r,s/2\}}\right)$; moreover, finite element methods of degree $\max \{r,s\}$ are minimal-error algorithms. We use a Picard method to approximate the solution of the resulting linear systems, since Gaussian elimination will be too expensive. We find that the total ε-complexity of the problem is at least $\Omega \left({\epsilon }^{-1/\min \{r,s/2\}}\right)$ and at most $O({\epsilon }^{-1/\min \{r,s/2\}}\ln {\epsilon }^{-1})$, the upper bound being attained by using $O(\ln {\epsilon }^{-1})$ Picard iterations.