# A preconditioned GMRes method for non-SPD problems

## Dipl.-Ing. Michael Kolmbauer

April 22, 2009, 3:30 p.m. T 212

Parabolic initial-boundary value problems with harmonic excitations lead to complex elliptic problems in the time-domain. The ﬁnite element discretization of the real reformulation results in a system of
the form $A_N u = f$ with $A_N = A + N$ , where $A$ is SPD and $N$ is non SPD. In fact, in our case, $N$ is block-skew-symmetric. In this talk, we review the paper ”A Preconditioned GMRes Method for Nonsymmetric or Indeﬁnite Problems” by J. Xu and X.-C. Cai (Mathematics of Computation, Vol. 59, No. 200, 1992, pp. 311-319) that provides a general framework for constructing a two level preconditioner $B_N$ for $A_N$.

It is shown, that an optimal preconditioner can be constructed by a coarse mesh solver plus any other optimal preconditioner $B$ designed for the SPD part $A$. The resultant operator $B_N A_N$ can be estimated in terms of the condition number of its symmetric part $BA$, and therefore the GMREs method applied to the non symmetric or indeﬁnite problem is optimal.

In this talk the abstract theory of the mentioned preconditioner is presented and the convergence rate estimations of the corresponding GMRes method are proven.