Bi-CG: An effective solver for three fields domain decomposition method in parallel environments

After applying substructuring preconditioner for the linear system stemming from the three fields domain decomposition method for elliptic boundary value problems, the preconditioned system will be nonsymmetric and Bi-Conjugate Gradient (Bi-CG) method can be applied. In this paper, we will show that in parallel environments the Bi-CG method works as Preconditioned Conjugate Gradient (PCG) method which applied in Bertoluzza [Substructuring preconditioners for the three fields domain decomposition method, Math. Comp. 73(246) (2003) 659-689] and in this case extensive numerical tests, performed on both conforming and nonconforming formulation, show that Bi-CG can work faster than PCG.