Iterative and direct solvers for interface problems with lagrange multipliers

Special purpose iterative and direct solvers are proposed for solving nonpositive definite symmetric linear systems arising from the three-field hybrid variational principle, which enforces compatibility between independently modeled substructures in the weak sense. The basic idea of the proposed family of solvers is to transform a nonpositive definite linear system into an equivalent positive definite system, which can then be solved by either the iterative or direct method without pivoting. The two-level iterative approach proposed in this paper consists of resolving the lower frequency response of the source problem by means of the equivalent positive definite collocation problem and capturing higher frequency response using projected conjugate gradient method. Numerical experiments in 2-D and shells indicate that the total CPU time of the two-level iterative process is less than 20% higher than that of a corresponding collocation problem, but the benefit from increased accuracy and modeling flexibility clearly overshadows this increased cost.