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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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...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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...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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...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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...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]....

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...to investigate the robustness of the developed new schemes, we now consider the Helmholtz equation with varying wave numbers. It should be pointed out that although the problem has been considered in =-=[8, 11, 18, 13, 31]-=-, most of the theoretical or numerical studies in the Helmholtz equation are limited to constant wave numbers. In the model problem (1)-(3), the constant wave number k suggests that the wave propagate...