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Where does smoothness count the most for two-point boundary-value problems?

Werschulz, Arthur G.

We are concerned with the complexity of 2mth order elliptic two-point boundary-value problems Lu = f. Previous work on the complexity of these problems has generally assumed that we had partial information about the right- hand side f and complete information about the coefficients of L, often making unrealistic assumptions about the smoothness of the coefficients of L. In this paper, we study problems in which f has r derivatives in theLp-sense and for L having the usual divergence form
Lv = (−1)iDi(ai,jDjv), 0≤i,j≤m
with ai,j being ri,j -times continuously differentiable. We find that if continuous information is permissible, then theε-complexity is proportional to(1/ε)1/(r ̃+m), where
r ̃=min{r, min {ri,j −i}}, 0≤i,j≤m
and show that a finite element method (FEM) is optimal. If only standard information (consisting of function and/or derivative evaluations) is available, we find that the complexity is proportional to (1/ε)1/rmin , where
rmin = min{ r, min {ri,j } }, 0≤i,j≤m
and we show that a modified FEM (which uses only function evaluations, and not derivatives) is optimal.



More About This Work

Academic Units
Computer Science
Department of Computer Science, Columbia University
Columbia University Computer Science Technical Reports, CUCS-026-97
Published Here
April 25, 2011