We examine the Kac-Schwarz problem of specification of point in Grassmannian in the restricted case of gap-one first-order differential Kac-Schwarz operators. While the pair of constraints satisfying [K1,W ] = 1 always leads to Kontsevich type models, in the case of [K1,W ] = W the corresponding KP τ -functions are represented as more sophisticated matrix integrals.

We demonstrate that a small modification of the multiplicative, additive and restricted additive Schwarz preconditioner at the algebraic level, motivated by optimized Schwarz methods defined at the continuous level, leads to a significant reduction in the iteration count of the iterative solver. Numerical experiments using finite difference and spectral element discretiza-tions of the positive ...

We propose a non-overlapping Schwarz algorithm for solving “Discrete Duality Finite Volume” schemes (DDFV for short) on general meshes. In order to handle this problem, the first step is to propose and study a convenient DDFV scheme for anisotropic elliptic problems with mixed Dirichlet/Fourier boundary conditions. Then, we are able to build the corresponding Schwarz algorithm and to prove its ...

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...

This paper deals with the coupling between one-dimensional heat and wave equations in unbounded subdomains, as a simplified prototype of fluid-structure interaction problems. First we apply appropriate artificial boundary conditions that yield an equivalent problem, but bounded carry out stability analysis for this coupled problem truncated domains. Then devise optimized Schwarz-in-time (or Sch...

We provide a maximum norm analysis of a finite element Schwarz alternating method for a nonlinear elliptic boundary value problem of the form −Δu = f(u), on two overlapping sub domains with non matching grids. We consider a domain which is the union of two overlapping sub domains where each sub domain has its own independently generated grid. The two meshes being mutually independent on the ove...

Optimized Schwarz methods form a class of domain decomposition algorithms in which the transmission conditions are optimized in order to achieve fast convergence. They are usually derived for a model problem with two subdomains, and give efficient transmission conditions for the local coupling between neighboring subdomains. However, when using a large number of subdomains, a coarse space corre...

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...