[petsc-users] Krylov Method Takes Too Long to Solve

[Jie Cheng]

Hi 

I’m implementing finite element method on nonlinear solid mechanics. The main portion of my code that involves PETSc is that in each step, the tangent stiffness matrix A is formed and the increment of the nodal degrees of freedom is solved. Standard Newton’s iteration. The problem is: when I use Krylov methods to solve the linear system, the KSPsolve process takes too long, although only 2 or 3 iterations are needed. 

The finite element formulation is the displacement/pressure mixed formulation, which I believe is symmetric and positive-definite. However if I pick conjugate gradient method with ICC preconditioned, PETSc gives me a -8 converged reason, which indicates a non-positive-definite matrix. After a couple of trials and errors, the only pair that works is GMRES plus PCKSP. But as I said, the KSPsolve function takes too much time.

A typical problem I’m trying has 16906 rows and 16906 cols. The message printed out by ksp_view is as following:

KSP Object: 1 MPI processes
  type: gmres
    GMRES: restart=30, using Classical (unmodified) Gram-Schmidt Orthogonalization with no iterative refinement
    GMRES: happy breakdown tolerance 1e-30
  maximum iterations=10000, initial guess is zero
  tolerances:  relative=1e-05, absolute=1e-50, divergence=10000
  left preconditioning
  using PRECONDITIONED norm type for convergence test
PC Object: 1 MPI processes
  type: ksp
  KSP and PC on KSP preconditioner follow
  ---------------------------------
    KSP Object:    (ksp_)     1 MPI processes
      type: gmres
        GMRES: restart=30, using Classical (unmodified) Gram-Schmidt Orthogonalization with no iterative refinement
        GMRES: happy breakdown tolerance 1e-30
      maximum iterations=10000, initial guess is zero
      tolerances:  relative=1e-05, absolute=1e-50, divergence=10000
      left preconditioning
      using PRECONDITIONED norm type for convergence test
    PC Object:    (ksp_)     1 MPI processes
      type: ilu
        ILU: out-of-place factorization
        0 levels of fill
        tolerance for zero pivot 2.22045e-14
        matrix ordering: natural
        factor fill ratio given 1, needed 1
          Factored matrix follows:
            Mat Object:             1 MPI processes
              type: seqaij
              rows=16906, cols=16906
              package used to perform factorization: petsc
              total: nonzeros=988540, allocated nonzeros=988540
              total number of mallocs used during MatSetValues calls =0
                using I-node routines: found 7582 nodes, limit used is 5
      linear system matrix = precond matrix:
      Mat Object:       1 MPI processes
        type: seqaij
        rows=16906, cols=16906
        total: nonzeros=988540, allocated nonzeros=988540
        total number of mallocs used during MatSetValues calls =0
          using I-node routines: found 7582 nodes, limit used is 5
  ---------------------------------
  linear system matrix = precond matrix:
  Mat Object:   1 MPI processes
    type: seqaij
    rows=16906, cols=16906
    total: nonzeros=988540, allocated nonzeros=988540
    total number of mallocs used during MatSetValues calls =0
      using I-node routines: found 7582 nodes, limit used is 5

Could anyone give me any suggestion please?

Thanks
Jie Cheng
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