1,791 research outputs found

    Optimized Schwarz Methods for Maxwell equations

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    Over the last two decades, classical Schwarz methods have been extended to systems of hyperbolic partial differential equations, and it was observed that the classical Schwarz method can be convergent even without overlap in certain cases. This is in strong contrast to the behavior of classical Schwarz methods applied to elliptic problems, for which overlap is essential for convergence. Over the last decade, optimized Schwarz methods have been developed for elliptic partial differential equations. These methods use more effective transmission conditions between subdomains, and are also convergent without overlap for elliptic problems. We show here why the classical Schwarz method applied to the hyperbolic problem converges without overlap for Maxwell's equations. The reason is that the method is equivalent to a simple optimized Schwarz method for an equivalent elliptic problem. Using this link, we show how to develop more efficient Schwarz methods than the classical ones for the Maxwell's equations. We illustrate our findings with numerical results

    Domain Decomposition preconditioning for high-frequency Helmholtz problems with absorption

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    In this paper we give new results on domain decomposition preconditioners for GMRES when computing piecewise-linear finite-element approximations of the Helmholtz equation Δu(k2+iε)u=f-\Delta u - (k^2+ {\rm i} \varepsilon)u = f, with absorption parameter εR\varepsilon \in \mathbb{R}. Multigrid approximations of this equation with ε0\varepsilon \not= 0 are commonly used as preconditioners for the pure Helmholtz case (ε=0\varepsilon = 0). However a rigorous theory for such (so-called "shifted Laplace") preconditioners, either for the pure Helmholtz equation, or even the absorptive equation (ε0\varepsilon \not=0), is still missing. We present a new theory for the absorptive equation that provides rates of convergence for (left- or right-) preconditioned GMRES, via estimates of the norm and field of values of the preconditioned matrix. This theory uses a kk- and ε\varepsilon-explicit coercivity result for the underlying sesquilinear form and shows, for example, that if εk2|\varepsilon|\sim k^2, then classical overlapping additive Schwarz will perform optimally for the absorptive problem, provided the subdomain and coarse mesh diameters are carefully chosen. Extensive numerical experiments are given that support the theoretical results. The theory for the absorptive case gives insight into how its domain decomposition approximations perform as preconditioners for the pure Helmholtz case ε=0\varepsilon = 0. At the end of the paper we propose a (scalable) multilevel preconditioner for the pure Helmholtz problem that has an empirical computation time complexity of about O(n4/3)\mathcal{O}(n^{4/3}) for solving finite element systems of size n=O(k3)n=\mathcal{O}(k^3), where we have chosen the mesh diameter hk3/2h \sim k^{-3/2} to avoid the pollution effect. Experiments on problems with hk1h\sim k^{-1}, i.e. a fixed number of grid points per wavelength, are also given

    New optimized Schwarz algorithms for one dimensional Schr\"odinger equation with general potential

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    The aim of this paper is to develop new optimized Schwarz algorithms for the one dimensional Schr{\"o}dinger equation with linear or nonlinear potential. After presenting the classical algorithm which is an iterative process, we propose a new algorithm for the Schr{\"o}dinger equation with time-independent linear potential. Thanks to two main ingredients (constructing explicitly the interface problem and using a direct method on the interface problem), the new algorithm turns to be a direct process. Thus, it is free to choose the transmission condition. Concerning the case of time-dependent linear potential or nonlinear potential, we propose to use a pre-processed linear operator as preconditioner which leads to a preconditioned algorithm. Numerically , the convergence is also independent of the transmission condition. In addition, both of these new algorithms implemented in parallel cluster are robust, scalable up to 256 sub domains (MPI process) and take much less computation time than the classical one, especially for the nonlinear case

    A rapidly converging domain decomposition method for the Helmholtz equation

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    A new domain decomposition method is introduced for the heterogeneous 2-D and 3-D Helmholtz equations. Transmission conditions based on the perfectly matched layer (PML) are derived that avoid artificial reflections and match incoming and outgoing waves at the subdomain interfaces. We focus on a subdivision of the rectangular domain into many thin subdomains along one of the axes, in combination with a certain ordering for solving the subdomain problems and a GMRES outer iteration. When combined with multifrontal methods, the solver has near-linear cost in examples, due to very small iteration numbers that are essentially independent of problem size and number of subdomains. It is to our knowledge only the second method with this property next to the moving PML sweeping method.Comment: 16 pages, 3 figures, 6 tables - v2 accepted for publication in the Journal of Computational Physic
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