Two-level block preconditioners for contact problems

Contact mechanics can be addressed numerically by Finite Elements using either a penalty formulation or the Lagrange multipliers. The penalty approach leads to a linearized symmetric positive definite system which can prove severely ill-conditioned, with the iterative solution to large 3D problems requiring expensive preconditioners to accelerate, or even to allow for, convergence. If the nodal unknowns are numbered properly, the system matrix takes on a two-level block structure that may be efficiently preconditioned by matrices having the same block structure. The present study addresses two different approaches, the Mixed Constraint Preconditioner (MCP) and the Multilevel Incomplete Factorization (MIF). It is shown that both MCP and MIF can prove very effective in the solution of large size 3D contact problems discretized by a penalty formulation, where classical algebraic preconditioners, such as the incomplete Cholesky decomposition, may exhibit poor performances.