A note on the complexity of solving Poisson's equation for spaces of bounded mixed derivatives
We study the complexity of solving the d-dimensional Poisson equation on ]0, 1[(d). We restrict ourselves to cases where the solution ii lies in some space of functions of bounded mixed derivatives (with respect to the L-infinity- or the L-2-norm) up to partial derivative(2d).Pi(j=1)(d) partial derivative x(j)(2). An upper bound for the: complexity of computing a solution of some prescribed accuracy epsilon with respect to the energy norm is given, which is proportional to epsilon(-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(epsilon(-1)) is gn-en. and. second, we provide a practically usable hierarchical basis finite element method of this complexity O(epsilon(-1)), i.e,, without logarithmic terms growing exponentially in ii. at least for our sparse grid setting with its underlying smoothness requirements. (C) 1999 Academic Press.
complexity poisson equation finite element method sparse grids multilevel methods