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I'm implementing a solver of Finite Element Method, and to solve the linear system of equations I'm using gmres from MKL of Intel. Exists the option with and without a preconditioning. In what case it is not necessary the preconditioning?
The matrix is sparse and not symmetric.
In my experience, you always need (or better use) some form of preconditioning. The type and complexity of the precondition would vary depending on the task though.
From Y. Saad, Iterative Methods for Sparse Linear Systems:
Preconditioning is a key ingredient for the success of Krylov subspace methods...
Lack of robustness is a widely recognized weakness of iterative solvers, relative to direct solvers. This drawback hampers the acceptance of iterative methods in industrial applications despite their intrinsic appeal for very large linear systems. Both the efficiency and robustness of iterative techniques can be improved by using preconditioning. (...) In general, the reliability of iterative techniques, when dealing with various applications, depends much more on the quality of the preconditioner than on the particular Krylov subspace accelerators used.
The simplest form of preconditioning is Jacobi or diagonal preconditioner. Such preconditioners are extremely quick to calculate and apply to your matrix. It is usually the minimum type of preconditioning that one would ever use for a practical problem.
So, I would say one should always go with at least diagonal preconditioning for the iterative solver. I often heard this phrase (don't know whom to attribute to):
Diagonal preconditioner is always cheap. Sometimes, effective.
From the same Y. Saad book:
In addition, it has been observed that for “easy problems,” the reduced system can often be solved efficiently with only diagonal preconditioning.
However, if we assume that the matrix for the linear system is diagonally dominant and elements on the diagonal are very close to each other, then probably even diagonal preconditioning is not going to improve things a lot.
The only practical application I had found, is to use a run without a preconditioner in order to compare its performance with other preconditioners. However, even in this case, I happened to run into research papers with slightly ambiguous labels in figures and tables: "No preconditioner (Jacobi)" or "No preconditioner" implying that only a diagonal preconditioner is used (which I found later trying to reproduce the results).
Note, there is a wide variety of literature on the preconditioning techniques including many questions in this community: preconditioning; however, I limited my answer towards a no-preconditioner aspect.
