abstract:
In [*Math. Comp*, 70 (2001), 27-75] and
[*Found. Comput. Math.*, 2(3) (2002), 203-245], Cohen, Dahmen and
DeVore introduced adaptive wavelet methods for solving operator
equations. These papers meant a break-through in the field, because
their adaptive methods were not only proven to converge, but also with
a rate better than that of their non-adaptive counterparts in cases
where the latter methods converge with a reduced rate due to a lacking
regularity of the solution. Until then, adaptive methods were usually
assumed to converge via a saturation assumption. An exception was given
by the work of Dörfler in [*SIAM J. Numer. Anal.*, 33 (1996),
1106-1124], where an adaptive finite element method was proven to
converge, with no rate though.
In the first talk, I will give a detailed description of the adaptive
wavelet methods, and present their convergence analysis. In the second
talk, some of the recent development in the field will be discussed, such as
the application of the schemes to tensor product wavelet bases, the
design of special wavelets that lead to truly sparse stiffness matrices,
and the adaptive wavelet solution of parabolic evolution equations. |