The Complexity of Two-Point Boundary-Value Problems with Analytic Data

Abstract

Previous work on the complexity of elliptic boundary-value problems Lu = f assumed that class F of problem elements f was the unit ball of a Sobolev space. In this paper, we assume that F consists of analytic functions. To be specific, we consider the ϵ-complexity of a model two-point boundary-value problem −u″ + u = f in I = (−1, 1) with natural boundary conditions u′(−1) = u′(1) = 0, and the class F consists of analytic functions f bounded by 1 on a disk of radius ρ ≥ 1 centered at the origin. We find that if ρ > 1, then the ϵ-complexity is Θ(ln(ϵ−1)) as ϵ → 0, and there is a finite element p-method (in the sense of Babuška) whose cost is optimal to within a constant factor. If ρ = 1, we find that the ϵ-complexity is Θ(ln2(ϵ−1)) as ϵ → 0, and there is a finite element (h, p)-method whose cost is optimal to within a constant factor.