Conjugate gradient method for dual-dual mixed formulation
Gabriel N. Gatica, Norbert Heuer
Math. Comp. 71 (240), 1455-1472, 2002.
We deal with the iterative solution of linear systems arising
from so-called dual-dual mixed finite element formulations.
The linear systems are of a two-fold saddle point structure;
they are indefinite and ill-conditioned.
We define a special inner product that makes matrices
of the two-fold saddle point structure, after a specific
transformation, symmetric and positive definite. Therefore,
the conjugate gradient method with this special inner product
can be used as iterative solver.
For a model problem, we prove that simple scalings lead to
a solution procedure which requires O(h^{-1}) iterations.
Here, h is the mesh size of the underlying dual-dual mixed method.
Moreover, we propose a preconditioner which leads to a bounded
number of iterations. Numerical experiments for our model problem
confirming the theoretical results are also reported.