In this contribution we use the algebraic representation recently developed for the classical additive and multiplicative Schwarz methods in [FS99, BFNS01] to analyze the restrictive additive Schwarz (RAS) and restrictive multiplicative Schwarz (RMS) methods; see [CS96, CFS98, CS99, QV99]. RAS was introduced in [CS99] as an efficient alternative to the classical additive Schwarz preconditioner....

Numerical experiments have shown that two-level Schwarz methods, for the solution of discrete elliptic problems, 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 va...

While the classical Schwarz method can only be used with overlap, optimized Schwarz methods can also be used without overlap, which can be an advantage when simulating heterogeneous problems, problems with jumping coefficients, or also for independent mesh generation per subdomain. The analysis of nonoverlapping optimized Schwarz methods has so far been restricted to the case of straight interf...

We consider the solution of linear systems of algebraic equations that arise from elliptic nite element problems We study a two level overlapping domain decomposition method that can be viewed as a combination of the additive and multiplicative Schwarz methods This method combines the advantages of the two methods It converges faster than the additive Schwarz algorithm and is more parallelizabl...

Optimized Schwarz methods are working like classical Schwarz methods, but they are exchanging physically more valuable information between subdomains and hence have better convergence behavior. The new transmission conditions include also derivative information, not just function values, and optimized Schwarz methods can be used without overlap. In this paper, we present a new optimized Schwarz...

Coarse spaces play a crucial role in making Schwarz type domain decomposition methods scalable with respect to the number of subdomains. In this paper, we consider coarse spaces based on a class of partition of unity (PU) for some domain decomposition methods including the classical overlapping Schwarz method and a hybrid Schwarz. PU has been used as a very powerful tool in the theoretical anal...

New reverses of the Schwarz, triangle and Bessel inequalities in inner product spaces are pointed out. These results complement the recent ones obtained by the author in the earlier paper [13]. Further, they are employed to establish new Grüss type inequalities. Finally, some natural integral inequalities are stated as well.

Domain decomposition methods in nite element applied aerodynamics provides a real speed-up of the convergence and good parallel scalability, even with the minimum overlap approach used here. Furthermore, a new variant of Restricted Additive Schwarz procedure is tested and shows a very attractive scalability property.