Dear Dr David Wells, Thank you for the explanation. However, this only satisfies me partially because the very next statement in that tutorial says:
*"We would have to resort to other iterative solvers instead, such as MinRes, SymmLQ, or GMRES, that can deal with indefinite systems. However, then the next problem immediately surfaces: due to the zero block, there are zeros on the diagonal and none of the usual, "simple" preconditioners (Jacobi, SSOR) will work as they require division by diagonal elements."* If the cause of the indefiniteness is not due to the eigen-value test, but rather due to the simple fact the pivot elements in the lower-right block is zero, then we have only *ONE* root cause of all difficulties. I am rephrasing the issues as follows (please correct me if my understanding is incorrect) "*Two difficulties* arise due to the *zero pivots* in the bottom-right block of the matrix. 1. Firstly, following a classical result from linear algebra, such matrices are indefinite and the conjugate gradient solver cannot be applied. We would have to resort to other iterative solvers instead, such as MinRes, SymmLQ, or GMRES, that can deal with indefinite systems. 2. Secondly, due to the zero block, there are zeros on the diagonal and none of the usual, "simple" preconditioners (Jacobi, SSOR) will work as they require division by diagonal elements" Regards, Krishna On Saturday, February 1, 2020 at 7:13:18 PM UTC, David Wells wrote: > > Hi Krishna, > > This is a classic linear algebra result - a symmetric matrix is > positive definite if and only if it has positive pivots. Since this > matrix has a zero block it does not even have a full set of pivots so > it cannot be positive definite. > > Thanks, > David Wells > > On Sat, Feb 1, 2020 at 9:49 AM Krishnakumar Gopalakrishnan > <kris...@vt.edu <javascript:>> wrote: > > > > I realize that this question is not exactly about the code/concepts > behind deal.II library itself, but rather about a mathematical statement > from step-20. > > > > "After assembling the linear system we are faced with the task of > solving it. The problem here is: the matrix has a zero block at the bottom > right (there is no term in the bilinear form that couples the pressure p > with the pressure test function q), and it is indefinite". > > > > My question is about the indefiniteness of the matrix. In my > understanding, a matrix is indefinite if it has both positive and negative > eigen values. However, there has not been any discussion thus far in > step-20 that comment about the eigen-values. How was the statement about > the indefiniteness then made? Can someone here explain why that matrix is > indefinite? > > > > Regards, > > Krishna > > > > -- > > The deal.II project is located at http://www.dealii.org/ > > For mailing list/forum options, see > https://groups.google.com/d/forum/dealii?hl=en > > --- > > You received this message because you are subscribed to the Google > Groups "deal.II User Group" group. > > To unsubscribe from this group and stop receiving emails from it, send > an email to dea...@googlegroups.com <javascript:>. > > To view this discussion on the web visit > https://groups.google.com/d/msgid/dealii/ec88d69a-4b26-43cb-915b-b3ae1fdfa3aa%40googlegroups.com. > > > -- The deal.II project is located at http://www.dealii.org/ For mailing list/forum options, see https://groups.google.com/d/forum/dealii?hl=en --- You received this message because you are subscribed to the Google Groups "deal.II User Group" group. To unsubscribe from this group and stop receiving emails from it, send an email to dealii+unsubscr...@googlegroups.com. To view this discussion on the web visit https://groups.google.com/d/msgid/dealii/2e309994-0d67-4809-bb28-0c07a142184e%40googlegroups.com.
