[petsc-users] Symmetric non-positive definite system solved with CG+ML - why does it not fail?

[Jozsef Bakosi]

Hi folks,

I have a matrix as a result of a finite element discretization of the Poisson
operator and associated right hand side. As it turns out, the matrix is
symmetric but not positive definite since it has at least two negative small
eigenvalues. I have been solving this system without problem using the conjugate
gradients (CG) algorithm with ML as a preconditioner, but I'm wondering why it
works.

Shouldn't CG fail for a non-positive-definite matrix? Does PETSc do some
additional magic if it detects that the dot-product in the CG algorithm is
negative? Does it solve the system using the normal equations, A'A, instead?

Please let me know if I should send the matrix + rhs.

Thanks,
Jozsef