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we are using a first order implicit finite volume code for simulation of incompressible flows. At its core, the code utilizes a (non-preconditioned) GMRES method for solving linear systems given in webpage

In general, it is quite difficult to find an "ideal" setup of the GMRES algorithm for a given problem, i.e. especially to find a suitable size of the maximum Krylov space dimension (inner iterations). I wonder if there are any estimates for determining such parameters beforehand, in order to circumvent some of the trial and error procedure (which is really cumbersome and painful for large problems). Maybe someone can point me to literature where such an issue might be discussed....?

Is there any sparse matrix solver better or other than GMRES?

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This is all wasted effort if you are not using a preconditioner because no choice of parameters will lead to a robust or scalable algorithm, and the least-bad choice will be effected by preconditioning. After you have found a good preconditioner, you can look at the effect of restarts and decide on how to manage restarts in GMRES, as well as comparing to other methods such as BiCGStab and related.

See also the Embree reference cited here.

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  • $\begingroup$ Thank you very much for your answer Sir, Can you suggest me any good reference to read basics on Krylov space and GMRES? because I am just using program which is written by other person and now I got interest in learning this solver. Thanks. $\endgroup$ – Shri Apr 26 '13 at 4:33
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    $\begingroup$ @Shri, take a look at Yousef Saad's book on iterative solvers for sparse matrices. The book is freely available from Saad's web site and discusses in detail Krylov subspace methods (and GMRES, which he developed) and preconditioning. $\endgroup$ – Geoff Oxberry Apr 29 '13 at 9:49
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In addition to @jed-brown's answer, I can highly recommend the book Finite elements and fast iterative solvers: with applications in incompressible fluid dynamics by Elman, Silvester and Wathen. The book covers finite elements, but the general strategies for Krylov subspace methods also hold for finite volumes, of course.

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In addition to preconditioners mentioned by Jed Brown, there are adaptive strategies for selecting the restarting parameter, which also have the potential to speed up convergence. See for example A.H. Baker, E.R. Jessup, Tz.V. Kolev — A simple strategy for varying the restart parameter in GMRES(m).

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