A Schur complement system is a system of interface variables obtained from non-overlapping domain decomposition and static condensation. In linear elasticity problems, this system preserves symmetric, positive definite of an original system formulated by finite element method. Hence, Krylov subspace method, which is one of the single-level iterative solvers, and incomplete factorization method, which is one of the single-level preconditioners, can be applied directly to the Schur complement system likewise in the single domains case. But Krylov subspace method with incomplete factorization method shows h-dependence which means convergence properties affected by the size of elements and d-dependence affected by the size of subdomains.
Multigrid method, which is one of the multi-level iterative solvers, is known to have near h-independence convergence properties in single domain case. These convergence properties are caused by two main procedures of multigrid methods which selectively reduce the error modes. One procedure is smoothing procedure that can employ the single-level iterative solvers and plays a role in reducing error modes corresponding to high frequency eigenvalues. The other procedure is coarse grid correction procedure that makes the auxiliary coarse grid space to approximately represent error modes of low frequency eigenvalues first and then accelerate the convergence.
Geometric multigrid method (GMG) constructs the coarse grid space with the geometric data and uses geometric smoothers, but it has difficulties of implementation in complex 3D problems. Algebraic multigrid method (AMG) uses algebraic smoothers and constructs the coarse grid space with only stiffness data, but it needs more computational efforts in constructing coarse grid space. Recently, aggregation multigrid method, which is another multigrid method, has been developed and shows efficient and robust performances in structural problems. This is a hybrid of AMG and GMG because it uses algebraic smoothers and coarse grid space based on the aggregation methods which use rigid body modes with geometric data.
In application of the multigrid methods to the Schur complement systems, the algebraic multigrid methods were simply adopted because of using stiffness data only. However, the aggregation multigrid methods which should use the geometric data lead to difficulties because the Schur complement matrices through condensation procedures are not directly connected to the geometric data. So, we use the indirect construction of coarse grid space with proposed three steps. First, the Schur complement system is transformed back to the original system using intermediate block diagonal system. Secondly, domainwise aggregation method is applied to the original system in order to maintain the relations between subdomains and interfaces. As the last step, the Schur complement system of the coarse grid space is constructed by condensation procedure. Through the three steps, we can obtain the relations of coarse/fine grid Schur complement systems.
Using the relations of the Schur complement systems, we propose the new multi-level preconditioner for the Schur complement system and theoretically prove that the convergence properties of multigrid method are also preserved in the Schur complement system. Various numerical experiments show that proposed preconditioner has not only the near h-independence but also the near d-independence convergence properties. In addition, merits of domain decomposition method using the interfacial system are well mixed with those of the aggregation multigrid method. Therefore, we acquire an efficient and robust iterative solver for the Schur complement system and it shows competitive results even in multi-loading cases.
