Optimized Schwarz methods are based on transmission conditions between subdomains which are optimized for the problem class that is being solved. Such optimizations have been performed for many different types of partial differential equations, but were almost exclusively based on the assumption of straight interfaces. We study in this paper the influence of curvature on the optimization, and w...

We study the additive and multiplicative Schwarz domain decomposition methods for elliptic boundary value problem of order 2r based on an appropriate spline space of smoothness r ? 1. The nite element method reduces an elliptic boundary value problem to a linear system of equations. It is well known that as the number of triangles in the underlying triangulation is increased, which is indispens...

The Schwarz problem for J-analytic functions in an ellipse is considered. In this case, the matrix J assumed to be two-dimensional with different eigenvalues located above real axis. reduced equivalent boundary value scalar functional equation depending on parameter l. This determined by Jordan basis of J. An analysis was performed. It shown that l∈[0,1], solution exists uniquely Hölder classes...

Abstract Curve shortening in the z -plane which, at a given point on curve, normal velocity of curve is equal to curvature, shown satisfy $$S_tS_z=S_{zz}$$ S t z = zz , where S ( t ) Schwarz funct...

In recent years, attention has been devoted to the development of efficient iterative solvers for the solution of the linear system of equations arising from the discontinuous Galerkin (DG) discretization of a range of model problems. In the framework of two level preconditioners, scalable non-overlapping Schwarz methods have been proposed and analyzed for the h–version of the DG method in the ...

For the solution of non-symmetric or indefinite linear systems arising from discretizations of elliptic problems, two-level additive Schwarz preconditioners are known to be optimal in the sense that convergence bounds for the preconditioned problem are independent of the mesh and the number of subdomains. These bounds are based on some kind of energy norm. However, in practice, iterative method...

We analyze the convergence of the multiplicative Schwarz method applied to nonsymmetric linear algebraic systems obtained from discretizations of one-dimensional singularly perturbed convection-diffusion equations by upwind and central finite differences on a Shishkin mesh. Using the algebraic structure of the Schwarz iteration matrices we derive bounds on the infinity norm of the error that ar...