Where Does Smoothness Count the Most for Two-Point Boundary-Value Problems?

✍ Scribed by Arthur G Werschulz

• Publisher: Elsevier Science
• Year: 1999
• Tongue: English
• Weight: 178 KB
• Volume: 15
• Category: Article
• ISSN: 0885-064X
• DOI: 10.1006/jcom.1999.0510

✦ Synopsis

We study the complexity of scalar 2m th order elliptic two-point boundary-value problems Lu=f, error being measured in the energy norm. 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. In this paper, we study the complexity of such problems when, in addition to partial information about f, we have only partial information about the coefficients of L. More precisely, we suppose that f has r derivatives in the L p -sense, with r &m and p # [2, ], and that L has the usual divergence form Lv= 0 i, j m (&1) i D i (a i, j D j v), with a i, j being r i, j -times continuously differentiable, where r i, j 0. We first suppose that continuous linear information is available. Let r~=min[r, min 0 i, j m [r i, j &i]]. If r~=&m, the problem is unsolvable; for r~>&m, we find that the =-complexity is proportional to (1Â=) 1Â(r~+m) , and we show that a finite element method (FEM) is optimal. We next suppose that only standard information (consisting of function andÂor derivative evaluations) is available. Let r min =min[r, min 0 i, j m [r i, j ]]. If r min =0, the problem is unsolvable; for r min >0, we find that the =-complexity is proportional to (1Â=) 1Âr min , and we show that a modified FEM (which uses only function evaluations, and not derivatives) is optimal.