In the past two decades, the FETI method introduced in [10] and its variants have become a class of popular methods for the parallel solution of large-scale finite element problems, see e.g. [11], [9], [14], [15], [8]. A key ingredient in this class of methods is a good preconditioner for the dual Schur complement system whose operator is a weighted sum of subdomain Neumann to Dirichlet (NtD) m...

For a Hilbert space H of functions let H H be the space of weak products of functions in H, i.e. all functions h that can be written as h = ∑∞ i=1 figi for some fi, gi ∈ H with ∑∞ i=1 ‖fi‖‖gi‖ <∞. Let D denote the Dirichlet space of the unit circle ∂D, i.e. the nontangential limits of functions f ∈ Hol(D) with ∫ D |f ′|2dA < ∞ and let Dh be the harmonic Dirichlet space, which consists of functi...

let $omega_x$ be a bounded, circular and strictly convex domain of a banach space $x$ and $mathcal{h}(omega_x)$ denote the space of all holomorphic functions defined on $omega_x$. the growth space $mathcal{a}^omega(omega_x)$ is the space of all $finmathcal{h}(omega_x)$ for which $$|f(x)|leqslant c omega(r_{omega_x}(x)),quad xin omega_x,$$ for some constant $c>0$, whenever $r_{omega_x}$ is the m...

We study connections between orthogonal polynomials, reproducing kernel functions, and polynomials p minimizing Dirichlet-type norms ‖pf − 1‖α for a given function f . For α ∈ [0, 1] (which includes the Hardy and Dirichlet spaces of the disk) and general f , we show that such extremal polynomials are non-vanishing in the closed unit disk. For negative α, the weighted Bergman space case, the ext...

This paper continues the study of the mixed problem for the Laplacian. We consider a bounded Lipschitz domain Ω ⊂ Rn, n ≥ 2, with boundary that is decomposed as ∂Ω = D ∪ N , D and N disjoint. We let Λ denote the boundary of D (relative to ∂Ω) and impose conditions on the dimension and shape of Λ and the sets N and D. Under these geometric criteria, we show that there exists p0 > 1 depending on ...

Considering difference equations in discrete space instead of differential equations in Euclidean space, we investigate a probabilistic formula for the solution of the Dirichlet problem for biharmonic functions. This formula involves the expectation of a weighted sum of the pay-offs at the successive times at which the Markov chain is in the complement of the domain. To make the infinite sum co...