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...en leads to very large sparse matrices A which are highly indefinite and ill-conditioned. It has been observed that, in the direct factorization of A the dense intermediate matrix may be compressible =-=[11, 25]-=-. We implemented a parallel multifrontal factorization method for such a matrix A based on the multifrontal method [9] with the nested dissection reordering [12]. In nested dissection, an n × n mesh i...

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...rid methods. We refer to the review articles Erlangga [15], Osei-Kuffuor and Saad [22] and the references therein for further information. Our work is inspired by the recent work of Engquist and Ying =-=[14]-=- in which a sweeping preconditioner is constructed by an approximate LDL t factorization which eliminates the unknowns layer by layer. The Schur complement matrix of the factorization is approximated ...

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...4305, USA. lexing@math.stanford.edu Article published by EDP Sciences c© EDP Sciences, SMAI 2014 434 P. TSUJI ET AL. degrade as the frequency is increased [9], and so several sweeping preconditioners =-=[6,7]-=- were recently introduced in order to alleviate this problem in the context of Helmholtz equations. Both of the original sweeping preconditioner approaches approximate a block LDLT factorization organ...

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...ω3), every linear solve required Ω(ω4) work with iterative techniques. Engquist and Ying recently introduced two classes of sweeping preconditioners for Helmholtz equations without internal resonance =-=[14, 15]-=-: Both approaches approximate a block LDLT factorization of the Helmholtz operator in block tridiagonal form in a manner which exploits a radiation boundary condition. The first approach performs a bl...

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...ach can be traced back to the AILU method of Gander and Nataf in 2001 [35, 37]. The first linear complexity claim was perhaps made in the work of Engquist and Ying on sweeping preconditioners in 2011 =-=[28, 29]-=- – a special kind of domain decomposition into grid-spacing-thin layers. In 2013, Stolk [65] restored the flexibilty to consider coarser layerings, with a domain decomposition method that realizes int...

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...ments of certain PDE discretization matrices, and they became a crucial component of modern Helmholtz preconditioners, such as Schwarz domain decomposition [12, 23] and the sweeping preconditioner in =-=[22]-=-. Preliminary results have shown successful application to a multilevel domain decomposition preconditioner, and a related Schlumberger patent application is pending. 20 V. DRUSKIN, S. GÜTTEL, AND L....

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...ized to N iterations with N subdomains, or one iteration when using an alternating sweep, and is the underlying mechanism for the good convergence of the sweeping preconditioner recently presented in =-=[16]-=-. Since the DtN operators are in general expensive, so-called optimized Schwarz waveform relaxation methods use local approximations; for a complete treatment of advection reaction diffusion equations...

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...α(x)ω Lc(x) , (1) where c(x) and α(x) are positive, smoothly varying coefficents, ω is the frequency parameter, and L is a typical length of the domain, which is inserted to make α dimensionless. See =-=[3, 11, 14, 23, 27, 12, 13]-=- for examples of preconditioners from the literature. For elliptic equations there are good preconditioning methods, such as multigrid and wavelet diagonal preconditioning [6, 8]. Their effect can be ...

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...itioner, the method uses the absorbing boundary condition (ABC), which is less effective compared to the PML, and hence the iteration count grows much more rapidly. Since the sweeping preconditioners =-=[6, 7]-=- were proposed, there have been a number of exciting developments for the numerical solutions of the high frequency Helmholtz equation, including but not limited to [15, 14, 18, 16, 17, 19, 2, 3, 20]....