We consider multilevel additive Schwarz methods with partial reenement. These algorithms are generalizations of the multilevel additive Schwarz methods developed by Dryja and Widlund and many others. We will give two diierent proofs by using quasi-interpolants under two diierent assumptions on selected reenement subregions to show that this class of methods has an optimal condition number. The ...

Quasi-hierarchical Powell-Sabin splines are C-continuous quadratic splines defined on a locally refined hierarchical triangulation. They admit a compact representation in a normalized B-spline basis. We prove that the quasi-hierarchical basis is in general weakly Lpstable, but for a broad class of hierarchical triangulations it is even strongly Lp-stable.

We study two kinds of quasi-interpolants (abbr. QI) in the space of C piecewise cubics in the plane, or in a rectangular domain, endowed with the highly symmetric triangulation generated by a uniform 6-direction mesh. It has been proved recently that this space is generated by the integer translates of two multi-box splines. One kind of QIs is of differential type and the other of discrete type...

Hierarchical Powell-Sabin splines are C-continuous piecewise quadratic polynomials defined on a hierarchical triangulation. The mesh is obtained by partitioning an initial conforming triangulation locally with a triadic split, so that it is no longer conforming. We propose a normalized quasi-hierarchical basis for this spline space. The B-spline basis functions have a local support, they form a...

In 1944 Favard [5, pp. 229, 239] introduced a discretely defined operator which is a discrete analogue of the familiar GaussWeierstrass singular convolution integral. In the present paper we consider a slight generalization Fn,σn of the Favard operator and its Durrmeyer variant F̃n,σn and study the local rate of convergence when applied to locally smooth functions. The main result consists of th...

In this piece of work, using three spatial grid points, we discuss a new two-level implicit cubic spline method of O(k + kh + h) for the solution of quasi-linear parabolic equation   , , , , xx x t u f x t u u u  , 0< x <1, t > 0 subject to appropriate initial and Dirichlet boundary conditions, where h > 0, k > 0 are grid sizes in space and time-directions, respectively. The cubic spline app...