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I am a graduate student majoring scientific computing. The numeric model I made caused a very ugly-looking saddle-point linear system. It is not symmetric at all and I will attach the sparsity pattern below. So far, I've only dealt with matrices with non-zero diagonal entries, so incomplete LU-type preconditioner with GMRES worked fine. But GMRES tends to converge slowly on this one and even worse, I cannot use any LU type preconditioners.

Could you recommend an iterative linear solver for this problem?

enter image description here

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  • $\begingroup$ LU can be used on matrices that have zero diagonal entries! $\endgroup$
    – Wolfgang Bangerth
    Commented Aug 20, 2019 at 22:37

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You should stick with GMRES, it is the only method that is essentially guaranteed to get a solution here. The real problem appears to be you need a better preconditioner. You could try sticking with LU but adding a diagonal mass matrix to the system with a constant multiplier that decreases with the linear residual of the system. This would allow you to get closer to the actual system as you solve your modified system. This is essentially a CFL ramping strategy for the linear system and is often used in CFD as part of a dual CFL system, and has some good results for pretty difficult to solve systems.

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  • $\begingroup$ It worked well! Thanks a lot. By the way, could you give me a reference about such preconditioning technique? I've tried but couldn't find one. $\endgroup$
    – Hoarsehinghing
    Commented Aug 29, 2019 at 5:18
  • $\begingroup$ In CFD you sometimes see it as a dual time stepping method. Looking that name up could help. $\endgroup$
    – BlaB
    Commented Sep 1, 2019 at 15:56
  • $\begingroup$ I will link a paper when I get back to my office I've been off on vacation. Sorry $\endgroup$
    – EMP
    Commented Sep 1, 2019 at 16:19
  • $\begingroup$ I also wouldn't say it's dual timestepping, it's dual CFL. Because we're using a second CFL condition to solve the linear problem rather than a second time step to solve the nonlinear one. $\endgroup$
    – EMP
    Commented Sep 1, 2019 at 19:02
  • $\begingroup$ @Hoarsehinghing the paper with an explanation fo the Dual CFL strategy that I'm familiar with is called: "Scalable Solution Strategies for Stabilized Finite-Element Flow Solvers on Unstructured Meshes, Part II" by Behzad Reza Ahrabi and Dimitri Mavriplis. Also, if you found my answer helpful, please consider accepting it as the answer. $\endgroup$
    – EMP
    Commented Sep 3, 2019 at 15:17

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