Sparse approximation of singularity functions
Andrej Nitsche (ETH Zürich)
We are concerned with the sparse approximation of functions of type
on the d-dimensional unit cube with parameters , , smooth g, a smooth cut-off
function , and a function of the remaining coordinates but |x|, possibly singular as well
(e. g. containing edge singularities). These functions arise e. g. from corners of domains in solutions to elliptic PDEs.
Usually, they deteriorate the rate of convergence of numerical algorithms to approximate these solutions.
We show, that functions of this type - for a range of covering elliptic singularities - can be approximated with
respect to the norm by sparse grid wavelet spaces , , of biorthogonal spline wavelets of degree
p essentially at the rate p:
where is a weighted Sobolev norm and .