A Note on the Complexity of Solving Poisson's Equation for Spaces of Bounded Mixed Derivatives

Abstract

We study the complexity of solving the d-dimensional Poisson equation on ]0, 1[d. We restrict ourselves to cases where the solution u lies in some space of functions of bounded mixed derivatives (with respect to the L∞- or the L2-norm) up to ∂2d/∏dj=1∂x2j. An upper bound for the complexity of computing a solution of some prescribed accuracy ε with respect to the energy norm is given, which is proportional to ε−1. We show this result in a constructive manner by proposing a finite element method in a special sparse grid space, which is obtained by an a priori grid optimization process based on the energy norm. Thus, the result of this paper is twofold: First, from a theoretical point of view concerning the complexity of solving elliptic PDEs, a strong tractability result of the order O(ε−1) is given, and, second, we provide a practically usable hierarchical basis finite element method of this complexity O(ε−1), i.e., without logarithmic terms growing exponentially in d, at least for our sparse grid setting with its underlying smoothness requirements.