The complexity of two-point boundary-value problems with piecewise analytic data (Q1342509)

The \(\varepsilon\)-complexity of two-point boundary-value problem is studied for the case when the class \(F\) of problem elements \(f\) is a class of piecewise analytic functions (in a previous paper of the author in J. Complexity 9, 154-170 (1993; Zbl 0773.65065) the class of analytic functions \(F\) was dealt with). The pieces of a piecewise analytic function belong to a common class of analytic functions with ``breathing room''. They are defined by the difference in how much we know about their breakpoints into three cases: (a) \(F\) consists of functions with a known location of breakpoints. Then the \(\varepsilon\)-complexity is proportional to \(\ell n(\varepsilon^{-1})\) and a finite element \(p\)- method is nearly optimal. (b) It is known only that there are \(b \geq 2\) breakpoints (their location is unknown), then the \(\varepsilon\)- complexity is proportional to \(b \varepsilon^{-1}\). (c) Neither the location nor the number of breakpoints are known, then the problem is unsolvable for \(\varepsilon < \sqrt 2\).