A class of preconditioned conjugate gradient methods for the solution of a mixed finite element discretization of the biharmonic operator

The homogeneous Dirichlet problem for the biharmonic operator is solved as the variational formulation of two coupled second‐order equations. The discretization by a mixed finite element model results in a set of linear equations whose coefficient matrix is sparse, symmetric but indefinite. We describe a class of preconditioned conjugate gradient methods for the numerical solution of this linear system. The precondition matrices correspond to incomplete factorizations of the coefficient matrix. The numerical results show a low computational complexity in both number of computer operations and demand of storage.