In this paper we describe a non-local moving frame along a curve of pure spinors in O(2m, 2m)/P , and its associated basis of differential invariants. We show that the space of differential invariants of Schwarzian-type define a Poisson submanifold of the spinor Geometric Poisson brackets. The resulting restriction is given by a decoupled system of KdV Poisson structures. We define a generaliza...

Numerical experiments have shown that two-level Schwarz methods often perform very well even if the overlap between neighboring subregions is quite small. This is true to an even greater extent for a related algorithm, due to Barry Smith, where a Schwarz algorithm is applied to the reduced linear system of equations that remains after that the variables interior to the subregions have been elim...

Keywords: Elliptic PDEs Schwarz alternating method Nonmatching grids Finite element L 1 – error estimate a b s t r a c t In this paper, we study a nonmatching grid finite element approximation of linear elliptic PDEs in the context of the Schwarz alternating domain decomposition.We show that the approximation converges optimally in the maximum norm, on each subdomain, making use of the geometri...

We show in this paper how to properly discretize optimized Schwarz methods for the time-harmonic Maxwell equations using a discontinuous Galerkin (DG) method. Due to the multiple traces between elements in the DG formulation, it is not clear a priori how the more sophisticated transmission conditions in optimized Schwarz methods should be discretized, and the most natural approach does not lead...

Geometric study of a second-order Fuchsian differential equation u′′ − q(x)u = 0, where q is rational in x, has been made via the Schwarz map as well as via the hyperbolic and the derived Schwarz maps ([SYY]). When the equation admits an irregular singularity, such a study was first made in [SY] treating the confluent hypergeometric equation and the Airy equation. In this paper, we study the hy...

Optimized Schwarz methods are a new class of Schwarz methods with greatly enhanced convergence properties. They converge uniformly faster than classical Schwarz methods and their convergence rates dare asymptotically much better than the convergence rates of classical Schwarz methods if the overlap is of the order of the mesh parameter, which is often the case in practical applications. They ac...

In recent years, much attention has been given to domain decomposition methods for solving linear elliptic problems that are based on a partitioning of the domain of the physical problem. More recently, a new class of Schwarz methods known as optimized Schwarz methods was introduced to improve the performance of the classical Schwarz methods. In this paper, we investigate the performance of thi...

A convergence analysis is presented for additive Schwarz iterations when applied to consistent singular systems of equations of the form Ax = b. The theory applies to singular M -matrices with one-dimensional null space and is applicable in particular to systems representing ergodic Markov chains, and to certain discretizations of partial differential equations. Additive Schwarz can be seen as ...

Krylov-type methods are widely used in order to accelerate the convergence of 10 Schwarz-type methods in the linear case. Authors in [2] have shown that they ac11 celerate without overhead cost the convergence speed of Schwarz methods for dif12 ferent types of transmission conditions. In the nonlinear context, the well-known 13 class of Newton-Krylov-Schwarz methods (cf. [5]) for steady-state p...