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A direct solver with O（N） complexity for integral equations on one-dimensional domains

Frontiers of Mathematics in China 2012年第2期7卷 217-247页

作者： Adrianna GILLMAN Patrick M. YOUNG Per-Gunnar MARTINSSON Department of Applied Mathematics University of Colorado at Boulder BoulderCO 80309-0526 USA

An algorithm for the direct inversion of the linear systems arising from NystrSm discretization of integral equations on one-dimensional domains is described. The method typically has O（N） complexity when applied to boundary integral equations （BIEs） in the plane with non-oscillatory kernels such as those associated with the Laplace and Stokes＇ equations. The scaling coefficient suppressed by the ＂big-O＂ notation depends logarithraically on the requested accuracy. The method can also be applied to BIEs with oscillatory kernels such as those associated with the Helmholtz and time-harmonic Maxwell equations; it is efficient at long and intermediate wave-lengths, but will eventually become prohibitively slow as the wave-length decreases. To achieve linear complexity, rank： deficiencies in the off-diagonal blocks of the coefficient matrix are exploited. The technique is conceptually related to the H- and H2-matrix arithmetic of Hackbusch and coworkers, and is closely related to previous work on Hierarchically Semi-Separable matrices.

关键词： direct solver integral equation fast direct solver boundary value problem boundary integral equation hierarchically semi-separable matrix MSC 65R20,65F05

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Linear System Solutions of the Navier-Stokes Equations with Application to Flow over a Backward-Facing Step

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Open Journal of Fluid Dynamics 2023年第3期13卷 133-143页

作者： Achraf Badahmane LMPA Université du Littoral C&otildete d’Opale Calais Cedex France

Many applications in fluid mechanics require the numerical solution of sequences of linear systems typically issued from finite element discretization of the Navier-Stokes equations. The resulting matrices then exhibit a saddle point structure. To achieve this task, a Newton-based root-finding algorithm is usually employed which in turn necessitates to solve a saddle point system at every Newton iteration. The involved linear systems being large scale and ill-conditioned, effective linear solvers must be implemented. Here, we develop and test several methods for solving the saddle point systems, considering in particular the LU factorization, as direct approach, and the preconditioned generalized minimal residual (ΡGMRES) solver, an iterative approach. We apply the various solvers within the root-finding algorithm for Flow over backward facing step systems. The particularity of Flow over backward facing step system is an interesting case for studying the performance and solution strategy of a turbulence model. In this case, the flow is subjected to a sudden increase of cross-sectional area, resulting in a separation of flow starting at the point of expansion, making the system of differential equations particularly stiff. We assess the performance of the direct and iterative solvers in terms of computational time, numbers of Newton iterations and time steps.

关键词： Navier-Stokes Equation ΡGMRES direct solver Schur Approach Preconditioner

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