The Complexity of Multivariate Elliptic Problems with Analytic Data

Let F be a class of functions deened on a d-dimensional domain. Our task is to compute H m-norm "-approximations to solutions of 2mth-order elliptic boundary-value problems Lu = f for a xed L and for f 2 F. We assume that the only information we can compute about f 2 F is the value of a nite number of continuous linear functionals of f, each evaluation having cost c(d). Previous work has assumed that F was the unit ball of a Sobolev space H r of xed smoothness r, and it was found that the complexity of computing an "-approximation was comp("; d) = (c(d)(1=") d=(r+m)). Since the exponent of 1=" depends on d, we see that the problem is intractable in 1=" for any such F of xed smoothness r. In this paper, we ask whether we can break intractability by letting F be the unit ball of a space of innnite smoothness. To be speciic, we let F be the unit ball of a Hardy space of analytic functions deened over a complex d-dimensional ball of radius greater than one. We then show that the problem is tractable in 1=". More precisely, we prove that comp("; d) = (c(d)(ln 1=") d), where the-constant depends on d. Since for any p > 0, there is a function K() such that comp("; d) c(d)K(d)(1=") p for suuciently small ", we see that the problem is tractable, with (minimal) exponent 0. Furthermore, we show how to construct a nite element p-method (in the sense of Babu ska) that can compute an "-approximation with cost (c(d)(ln 1=") d). Hence this nite element method is a nearly optimal complexity algorithm for d-dimensional elliptic problems with analytic data.