No abstract

Consider & regularly-elliptic 2m-th order bound&ry problem LtI. = I '4'ith I e Hr{O), r ~-m. In previous work. we showed tht the 6nite-element method (FE~) using piecewise polynomi&!s of degree It is &!Ymptotic&!ly optim&l when It ~ 2m-1 + r. In this paper. we show th&t the FE~ is not a.symptotic&lly optim&l when this inequ&lity is violated. However, there exists &n &lgorithm. the Traub-Wa.silkowski-Wotni&kowski spline Iilgorithm. which is &lways optim&l. Moreover, the error of the FE~ can be arbitrvily larger th&n the erT"Or of the spline &lgorithm. We &!so obtain 8. nec~ary &nd sufficient condition for & G&lerkin method (or & gener&Jized G&lerkin method) to be• & spline Iilgorithm.

We study the complexity of 2mth order definite elliptic problems Lu ϭ f (with homogeneous Dirichlet boundary conditions) over a d-dimensional domain ⍀, error being measured in the H m (⍀)-norm. The problem elements f belong to the unit ball of W r,p (⍀), where p ʦ [2, ȍ] and r Ͼ d/p. Information consists of (possibly adaptive) noisy evaluations of f or the coefficients of L. The absolute error in each noisy evaluation is at most ͳ. We find that the nth minimal radius for this problem is proportional to n Ϫr/d ϩ ͳ, and that a noisy finite element method with quadrature (FEMQ), which uses only function values, and not derivatives, is a minimal error algorithm. This noisy FEMQ can be efficiently implemented using multigrid techniques. Using these results, we find tight bounds on the -complexity (minimal cost of calculating an -approximation) for this problem, said bounds depending on the cost c(ͳ) of calculating a ͳ-noisy information value. As an example, if the cost of a ͳ-noisy evaluation is c(ͳ) ϭ ͳ Ϫs (for s Ͼ 0), then the complexity is proportional to (1/) d/rϩs .