Strong convergence estimates for pseudospectral methods applied to ordinary boundary value problems are derived. The results are also used for a convergence analysis of the Schwarz algorithm (a special domain decomposition technique). Different types of nodes (Chebyshev, Legendre nodes) are examined and compared.
We give a constructive proof that for any bounded domain of the class there exists a strongly regular family of boundary-fitted tetrahedral meshes. We adopt a refinement technique introduced by Křížek and modify it so that a refined mesh is again boundary-fitted. An alternative regularity criterion based on similarity with the Sommerville tetrahedron is used and shown to be equivalent to other standard criteria. The sequence of regularities during the refinement process is estimated from below...
In dimension one it is proved that the solution to a total variation-regularized least-squares problem is always a function which is "constant almost everywhere" , provided that the data are in a certain sense outside the range of the operator to be inverted. A similar, but weaker result is derived in dimension two.