Balancing domain decomposition

Abstract

SUMMARYThe Neumann-Neumann algorithm is known to be an efficient domain decomposition preconditioner with unstructured subdomains for iterative solution of finiteelement discretizations of difficult problems with strongly discontinuous coefficients (De Roeck and Le Tallec, 1991). However, this algorithm suffers from the need to solve in each iteration an inconsistent singular problem for every subdomain, and its convergence deteriorates with increasing number of subdomains due to the lack of a coarse problem to propagate the error globally. We show that the equilibrium conditions for the singular problems on subdomains lead to a simple and natural construction of a coarse problem. The construction is purely algebraic and applies also to systems such as those that arise in elasticity. A convergence bound independent of the number of subdomains is proved and results of computational tests are reported.

Numerical validation of an homogenized interface model

“…[16]. We will use the balancing Neumann-Neumann algorithm [19] which was intensively studied and proved to be robust, [16], [20] for elasticity problems. An alternative is to use a dual method of FETI type [21].…”

“…where the constant C is independent of the subdomains diameters H i , discretization steps h i , aspect ratios α i and elasticity coefficients [19], [16], [20].…”

“…In practice we will use a bounded domain Note that condition (19) implies that the function f is involved in the…”

Discontinuous Coarse Spaces for DD-Methods with Discontinuous Iterates

“…The use of a coarse space [16] is the only way to provide information from distant subdomains, as they enable global information transfer, ensuring scalability. In this respect, well known methods are the two level additive Schwarz method [3], and the FETI [13] and balancing Neumann-Neumann methods [12,4,14]. See also [11] for non-symmetric problems.…”

A multiscale preconditioner for stochastic mortar mixed finite elements

“…doi:10.1016/j.cma.2010. 10.015 Neumann-Neumann [28], balancing [27,14,35], or multigrid [43]. An advantage of the multiscale mortar formulation is that coarser mortar grids imply fewer interface iterations resulting in fewer number of subdomain solves.…”