Complexity for a class of elliptic ordinary integro-differential equations

Abstract

Consider the variational form of the ordinary integro-differential equation (OIDE)−u″+u+∫01q(⋅,y)u(y)dy=f on the unit interval I, subject to homogeneous Neumann boundary conditions. Here, f and q respectively belong to the unit ball of Hr(I) and the ball of radius M1 of Hs(I2), where M1∈[0,1). For ε>0, we want to compute ε-approximations for this problem, measuring error in the H1(I) sense in the worst case setting. Assuming that standard information is admissible, we find that the nth minimal error is Θ(n−min⁡{r,s/2}), so that the information ε-complexity is Θ(ε−1/min⁡{r,s/2}); 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 Ω(ε−1/min⁡{r,s/2}) and at most O(ε−1/min⁡{r,s/2}ln⁡ε−1), the upper bound being attained by using O(ln⁡ε−1) Picard iterations.