176 research outputs found
Cross-Points in Domain Decomposition Methods with a Finite Element Discretization
Non-overlapping domain decomposition methods necessarily have to exchange
Dirichlet and Neumann traces at interfaces in order to be able to converge to
the underlying mono-domain solution. Well known such non-overlapping methods
are the Dirichlet-Neumann method, the FETI and Neumann-Neumann methods, and
optimized Schwarz methods. For all these methods, cross-points in the domain
decomposition configuration where more than two subdomains meet do not pose any
problem at the continuous level, but care must be taken when the methods are
discretized. We show in this paper two possible approaches for the consistent
discretization of Neumann conditions at cross-points in a Finite Element
setting
A New Domain Decomposition Method for the Compressible Euler Equations
In this work we design a new domain decomposition method for the Euler
equations in 2 dimensions. The basis is the equivalence via the Smith
factorization with a third order scalar equation to whom we can apply an
algorithm inspired from the Robin-Robin preconditioner for the
convection-diffusion equation. Afterwards we translate it into an algorithm for
the initial system and prove that at the continuous level and for a
decomposition into 2 sub-domains, it converges in 2 iterations. This property
cannot be preserved strictly at discrete level and for arbitrary domain
decompositions but we still have numerical results which confirm a very good
stability with respect to the various parameters of the problem (mesh size,
Mach number, ....).Comment: Submitte
Effective transmission conditions for domain decomposition methods applied to the time-harmonic curl-curl Maxwell's equations
The time-harmonic Maxwell equations describe the propagation of electromagnetic waves and are therefore fundamental for the simulation of many modern devices we have become used to in everyday life. The numerical solution of these equations is hampered by two fundamental problems: first, in the high frequency regime, very fine meshes need to be used in order to avoid the pollution effect well known for the Helmholtz equation, and second the large scale systems obtained from the vector valued equations in three spatial dimensions need to be solved by iterative methods, since direct factorizations are not feasible any more at that scale. As for the Helmholtz equation, classical iterative methods applied to discretized Maxwell equations have severe convergence problems.We explain in this paper a family of domain decomposition methods based on well chosen transmission conditions. We show that all transmission conditions proposed so far in the literature, both for the first and second order formulation of Maxwell's equations, can be written and optimized in the common framework of optimized Schwarz methods, independently of the first or second order formulation one uses, and the performance of the corresponding algorithms is identical. We use a decomposition into transverse electric and transverse magnetic fields to describe these algorithms, which greatly simplifies the convergence analysis of the methods. We illustrate the performance of our algorithms with large scale numerical simulations
Robin Schwarz algorithm for the NICEM Method: the Pq finite element case
In Gander et al. [2004] we proposed a new non-conforming domain decomposition
paradigm, the New Interface Cement Equilibrated Mortar (NICEM) method, based on
Schwarz type methods that allows for the use of Robin interface conditions on
non-conforming grids. The error analysis was done for P1 finite elements, in 2D
and 3D. In this paper, we provide new numerical analysis results that allow to
extend this error analysis in 2D for piecewise polynomials of higher order and
also prove the convergence of the iterative algorithm in all these cases.Comment: arXiv admin note: substantial text overlap with arXiv:0705.028
Domain decomposition algorithms for the two dimensional nonlinear Schrödinger equation and simulation of Bose-Einstein condensates
International audienceIn this paper, we apply the optimized Schwarz method to the two dimensional nonlinear Schrödinger equation and extend this method to the simulation of Bose-Einstein condensates (Gross-Pitaevskii equation). We propose an extended version of the Schwartz method by introducing a preconditioned algorithm. The two algorithms are studied numerically. The experiments show that the preconditioned algorithm improves the convergence rate and reduces the computation time. In addition, the classical Robin condition and a newly constructed absorbing condition are used as transmission conditions
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