where represents the speed of the fluid and is related to viscoelastic forces. The boundary conditions are and . After discretization with central differences with grid size , the equations may be written as with . We consider the Jacobian matrix for with the parameter values , , and , evaluated in the trivial steady state solution.

We solved linear systems by GMRES with 30 iteration vectors. The method was restarted by Morgan's implicitly restarted GMRES [334] keeping the 15 smallest Ritz pairs in the basis until the residual norm satisfied with . We used Algorithm 11.4 for computing the eigenvalue nearest . We consider two cases. For Case 1, we used a fixed . For Case 2, we used two steps with and the remaining steps solved the correction equation (11.5). From the results in Table 11.2 and Figure 11.1, we can see linear convergence for Case 1. Since the linear systems are solved more accurately () than the speed of the eigenvalue solver, and converge to zero with the same speed. For Case 2, the convergence is linear for Steps 1 and 2, since is constant. From the third iteration on, converges quadratically to zero. The converge linearly in steps 1 and 2, then we have quadratic convergence on steps 3 and 4, and then linear convergence again with convergence ratio .

The residual norm has two terms. The first term decreases very rapidly when is close to an eigenvalue. The decrease of the second term (the residual of the linear system solver) is often much more difficult when the pole lies close to the spectrum. In practice, we may need to balance the number of outer iterations (the eigenvalue solver) and the number of inner iterations (the linear system solver) by selecting the most optimal . This comment is also related to the end of §11.2.3.