Waveform Transmission Method, a New Waveform-relaxation Based Algorithm to Solve Ordinary Differential Equations in Parallel. (arXiv:0911.1166v1 [math.NA])

Waveform Relaxation method (WR) is a beautiful algorithm to solve Ordinary
Differential Equations (ODEs). However, because of its poor convergence
capability, it was rarely used. Virtual Transmission Method (VTM) is a new
distributed algorithm to solve large sparse linear systems, with an appreciable
convergence speed. In this paper, by marrying WR with VTM, a new distributed
algorithm, named Waveform Transmission Method (WTM), was born. WTM has better
convergence capability than the traditional WR algorithms.

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Original source : http://arxiv.org/abs/0911.1166...

Selection of Corners for the BDDC Method. (arXiv:0911.1245v1 [math.NA])

The Balancing Domain Decomposition by Constraints (BDDC) method has evolved
quite fast since its introduction in 2003, as the primal counterpart to the
earlier FETI-DP method. Recent results have shown close connection of these
methods and theoretically supported equivalent rate of convergence. In both
methods, a fundamental role is played by the coarse space. Optimal choice of
constraints on continuity of the coarse space is still not a satisfactorily
solved problem. The usual basic choice is a 'minimal' set of coarse nodes
(sometimes called corners), that assures invertibility of local subdomain
problems and also of the global coarse problem. However, this set alone does
not suffice for optimal preconditioning in 3D. For this reason, continuity of
some generalized degrees of freedom, such as average values on edges or faces
of subdomains, have to be added. While theoretically correct, this approach
does not easily offer a flexible size of desired coarse problem. In our
contribution, we compare this approach with adding more coarse nodes into the
coarse problem, which is technically simpler and allows flexible setting of
desired approximation. Most attention is given to the basic set of corners, and
a new algorithm for its selection is proposed.

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Original source : http://arxiv.org/abs/0911.1245...