cg+amg converges slower than gmres/bicgstab+amg for SPD matrix?

Question

Consider the problem, Ax=b.

If A is a SPD matrix, I think cg(solver)+amg(preconditioner) should converge faster that gmres/bicgstab(solver)++amg(preconditioner). But the result I get is gmres/bicgstab(solver)++amg(preconditioner) is faster. Does this make sense?

I use FEM to solve a Poisson equation. If I use uniform mesh to discretize the PDE, the coefficient matrix A will be a SPD matrix. If I use nonuniform mesh to deiscretize the PDE, will the coefficient matrix A still be a SPD matrix? A will still be symmetric, but how can I prove it is positive definite?

amg predonditioner is best for SPD problem. Why? I have some non-definite problem, and can't get convergent with amg precondition. Why?

Comments

This isn't a FEniCS question. If you're using FEniCS, you should summarise what solver options you're using, which linear algebra backend, etc, and preferably post complete but compact code.

Answer 1

Does this make sense?

You should compare CG with FOM and MINRES with GMRES. These two couples do apparently different things. CG should behave the same as FOM without restarts assuming exact arithmetic. In floating-point arithmetic orthogonalization method (Arnoldi resp. Lanczos) suffers from loss of orthogonality which basically delays the convergence and the delay may be different for both methods. The same holds for MINRES and GMRES. Note also that particular implementation details may play a role.

A will still be symmetric, but how can I prove it is positive definite?

Using Poincare/Friedrichs inequality (depends on BCs). Generally, Laplacian is positive semi-definite on $H^1 \times H^1$.

Comments

What is FOM?

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Full orhthogonalization method.