At the heart of numerical weather prediction algorithms lie a Laplace and positive definite Helmholtz problems on the sphere [12]. Recently, there has been interest in using finite elements [2] and domain decomposition methods [1, 10]. The Schwarz iteration [7, 8, 9] and its variants [9, 4, 5, 6, 3, 11] are popular domain decomposition methods. In this paper, we introduce improved transmission ...

Newton-Krylov methods are potentially well suited for the implicit solution of nonlinear problems whenever it is unreasonable to compute or store a true Jacobian. Krylov-Schwarz iterative methods are well suited for the parallel implicit solution of multidimensional systems of boundary value problems that arise in CFD. They provide good data locality so that even a high-latency workstation netw...

Newton-Krylov methods are potentially well suited for the implicit solution of nonlinear problems whenever it is unreasonable to compute or store a true Jacobian. Krylov-Schwarz iterative methods are well suited for the parallel implicit solution of multidimensional systems of boundary value problems that arise in CFD. They provide good data locality so that even a high-latency workstation netw...

The original additive average Schwarz method and its variants were introduced for solving second order elliptic boundary value problems. In this paper, we extend the original idea to fourth order problems by designing a variant of the additive average Schwarz method for the plate bending problem using the nonconforming Morley finite element for the discretization. Like the original average meth...

The convergence rate of a numerical procedure based on Schwarz Alternating Method(SAM) for solving elliptic boundary value problems depends on the selection of the interface conditions applied on the interior boundaries of the overlapping subdomains. It has been observed that the Robin condition (mixed interface condition), controlled by a parameter, can optimize SAM’s convergence rate. In [7],...

To compute the electrical resistance ( ~ conformal modulus) of a polygonally shaped resistorcut from a sheet of uniform resistivity, it suffices to find a conformal map of the polygon onto arectangle. Constructing such a map requires the solution of a Schwarz-Christoffel parameter prob-lem. First we show by examples that this is practical numerically. Then we consider an inverse 704...

The classical isoperimetric problem seeks the least-area way to enclose a single region of prescribed volume. About 200 BC, Zenodorus argued that a circle is the least-perimeter enclosure of prescribed area in the plane (see [Heath 60]). In 1884, Schwarz [Schwarz 1884] proved by symmetrization that a sphere minimizes surface area for a given volume in R. Isoperimetric problems arise naturally i...

Here we shall introduce the concept of harmonic balls in subdomains of Rn, through a mean value property for a sub-class of harmonic functions on such domains. In the complex plane, and for analytic functions, a similar concept fails to exist due to the fact that analytic functions can not have prescribed data on the boundary. Nevertheless, a two-phase version of the problem does exists, and gi...

In this paper we study a special class of monodromy evolving deformations (MED), which represents Halphen’s quadratic system. In 1996, Chakravarty and Ablowitz [4] showed that a fifth-order equation (DH-V) ω 1 = ω2ω3 − ω1(ω2 + ω3) + φ , ω 2 = ω3ω1 − ω2(ω3 + ω1) + θ , ω 2 = ω3ω1 − ω2(ω3 + ω1)− φθ, (1) φ = ω1(θ − φ) − ω3(θ + φ), θ = −ω2(θ − φ)− ω3(θ + φ), which arises in complex Bianchi IX cosmol...

This paper studies the application of two-level Schwarz algorithms to several models of computational fluid dynamics. The purpose is to build an algorithm suitable for elliptic and convective models. The subdomain approximated solution relies on the incomplete lower-upper factorization. The algebraic coupling between the coarse grid and the Schwarz preconditioner is discussed. The deflation met...