Generalized Schwartz algorithm with variable parameters

Abstract

The paper introduces special classes of operators-reflection operators in the particle transport theory and Poincare -Steklov operators in elliptic boundary value problems. The properties of these operators and the general theory of iterative processes are used to propose techniques for construction and investigation of various algorithms of the generalized Schwartz method (the domain decomposition method). Estimates of the convergence rate for a number of specific algorithms are proved.Assume that // = D 2~1 C 2 D 1~1 C 1 , Sp(//) e [ra,Af], where 1 <£ [m,M], and m<M are spectral bounds. The system of eigenvalues of the operator Η is assumed to form a '

Convergence estimates for an higher order optimized Schwarz method for domains with an arbitrary interface

“…and a similar one forû (n) 2 . The convergence rate has been shown to be 1 − O(h 1/4 ) (see (4.39) in [13]) and the optimal parameters are p = O(h −1/4 ) and q = O(h 3/4 ) (see (4.30) in [13]).…”

“…Many authors have studied this method or its variations. In [2,3], the interface operator is expressed in terms of Poincare-Steklov operators. The optimal value of λ is shown to be O(h −1/2 ) which yields a spectral radius bounded above by 1 − O(h 1/2 ).…”

A numerical algorithm based on probing to find optimized transmission conditions

“…[2]), for the solution of the continuous Steklov-Poincaré equation. This interesting point of view has been discussed in [1,3]. At the discrete level, it results in the equivalence between a discretization of (2) and the ADI scheme…”

Convergence estimates for an optimized Schwarz method for PDEs with discontinuous coefficients

“…In [1] and [15], for the Poisson problem, the interface operator is expressed in terms of Poincaré-Steklov operators. The optimal value of λ is shown to be O(h −1/2 ) which yields a spectral radius bounded by 1 − O(h 1/2 ), where h is the discretization parameter along the interface.…”