The Complexity of the Poisson Problem for Spaces of Bounded Mixed Derivatives

We are interested in the complexity of the Poisson problem with homogeneous Dirichlet boundary conditions on the d-dimensional unit cube Ω. Error is measured in the energy norm, and only standard information (consisting of function evaluations) is available. In previous work on this problem, the standard assumption has been that the class F of problem elements has been the unit ball of a Sobolev space of fixed smoothness r, in which case the ϵ-complexity is proportional to ϵ to -d/r. Given this exponential dependence on d, the problem is intractable for such classes F. In this paper, we seek to overcome this intractability by allowing F to be the unit ball of a space Hp(Ω) of bounded mixed derivatives, with p a fixed multi-index with positive entries. We find that the complexity is proportional to c(d)(1/ϵ) 1/pmin[ln(1/ϵ)] and we give bounds on b = bp,d. Hence, the problem is tractable in 1/ϵ with exponent at most 1/ϵmin. The upper bound on the complexity (which is close to the lower bound) is attained by a modified finite element method (MFEM) using discrete blending spline spaces; we obtain an explicit bound (with no hidden constants) on the cost of using this MFEM to compute ϵ-approximations. Finally, we show that for any positive multi-index p, the Poisson problem is strongly tractable, and that the MFEM using discrete blended piecewise polynomial splines of degree p is a strongly polynomial time algorithm. In particular, for the case p=1, the MFEM using discrete blended piecewise linear splines produces an ϵ-approximation with cost at most 0.839262(c(d)+2)(1/ϵ)to 5.07911