1997 Reports

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.