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NUMERICAL STUDIES OF A CLASS OF COMPOSITE PRECONDITIONERS

Journal of Computational Mathematics 2014年第2期32卷 136-151页

作者： Qiang Niu Michael Ng Mathematics and Physics Centre Xi'an Jiaotong-Liverpool University Suzhou 215123 China Department of Mathematics Hong Kong Baptist University Kowloon Tong Hong Kong

In this paper, we study a composite preconditioner that combines the modified tangential frequency filtering decomposition with the ILU（O） factorization. Spectral property of the composite preconditioner is examined by the approach of Fourier analysis. We illustrate that condition number of the preconditioned matrix by the composite preconditioner is asymptotically bounded by O（hp -2/3） on a standard model problem. Performance of the composite preconditioner is compared with other preconditioners on several problems arising from the discretization of PDEs with discontinuous coefficients. Numerical results show that performance of the proposed composite preconditioner is superior to other relative preconditioners.

关键词： Preconditioner ILU Tangential frequency filtering decomposition gmres.

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Efficient Variable-Coefficient Finite-Volume Stokes Solvers

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Communications in Computational Physics 2014年第10期16卷 1263-1297页

作者： Mingchao Cai Andy Nonaka John B.Bell Boyce E.Griffith Aleksandar Donev Courant Institute of Mathematical Sciences New York UniversityNew YorkNY 10012USA. Center for Computational Sciences and Engineering Lawrence Berkeley National LaboratoryBerkeleyCA 94720USA. Leon H.Charney Division of Cardiology Department of MedicineNew York University School of MedicineNYUSA. Department of Mathematics University of North CarolinaChapel HillNC 27599USA

We investigate several robust preconditioners for solving the saddle-point linear systems that arise from spatial discretization of unsteady and steady variablecoefficient Stokes equations on a uniform staggered *** on the success of using the classical projection method as a preconditioner for the coupled velocitypressure system[***,***.,228(2009),pp.7565–7595],as well as established techniques for steady and unsteady Stokes flow in the finite-element literature,we construct preconditioners that employ independent generalized Helmholtz and Poisson solvers for the velocity and pressure *** demonstrate that only a single cycle of a standard geometric multigrid algorithm serves as an effective inexact solver for each of these *** to traditional wisdom,we find that the Stokes problem can be solved nearly as efficiently as the independent pressure and velocity subproblems,making the overall cost of solving the Stokes system comparable to the cost of classical projection or fractional step methods for incompressible flow,even for steady flow and in the presence of large density and viscosity *** of the five preconditioners considered here are found to be robust to gmres.restarts and to increasing problem size,making them suitable for large-scale *** work opens many possibilities for constructing novel unsplit temporal integrators for finite-volume spatial discretizations of the equations of low Mach and incompressible flow dynamics.

关键词： Stokes flow variable density variable viscosity saddle point problems projection method preconditioning gmres.

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CGS/gmres.k): AN ADAPTIVE PRECONDITIONED CGS ALGORITHM FOR NONSYMMETRIC LINEAR SYSTEMS

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Numerical Mathematics A Journal of Chinese Universities(English Series) 1998年第2期7卷 145-158页

作者：曹海燕李兴伟

Recently Y. Saad proposed a flexible inner-outer preconditioned gmres.algorithm for nonsymmetric linear systems [4]. Following their ideas, we suggest an adaptive preconditioned CGS method, called CGS/gmres.(k), in which the preconditioner is constructed in the iteration step of CGS, by several steps of gmres.k). Numerical experiments show that the residual of the outer iteration decreases rapidly. We also found the interesting residual behaviour of gmres.for the skewsymmetric linear system Ax = b, which gives a convergence result for restarted gmres.(k). For convenience, we discuss real systems.

关键词： Krylov subspace methods CGS gmres.

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