Numerical Methods for Saddle Point Problems arising in GeomechanicsDevelopments in Engineering Computational Technology

The finite element (FE) integration of the coupled consolidation equations requires the solution of symmetric 2x2 block linear systems with an indefinite saddle point coefficient matrix. Because of ill-conditioning, the repeated solution in time of the FE linear equations may be a major computational issue requiring ad hoc preconditioning strategies to guarantee the efficient convergence of Krylov subspace methods. The linear system to be solved at each time step can be generally written as Ax=b where A=[{A BT}{B -C}] In this we review a class of block preconditioners which have been proved very efficient in the acceleration of Krylov solvers in many field such as constrained optimization, the Stokes problem, to mention a few. The inexact constraint preconditioners (ICP) are based on efficient approximations PA of the (1,1) block A and PB of the resulting Schur complement matrix S = B PA-1 B\T + C. Bergamaschi et al. [1,2] have developed an MCP (mixed constraint preconditioner) which couples two explicit-implicit approximations of the inverse of K provided by an approximate inverse preconditioner such as AINV, and the IC preconditioner, respectively. They compared MCP with a variant called mixed block triangular preconditioner (MBTP). In this chapter, following the work in [3], we give a detailed spectral analysis of MCP and MBTP, showing that the convergence properties of these block preconditioners for iterative methods are strictly connected with the approximation introduced by PA and PB. It is found that the eigenvalues, most of which are complex ones, are well clustered around unity, provided that eigenvalues of PA-1A and PS-1S are too. The bounds on real and complex eigenvalues are verified on a small test case. A number of test cases from moderate to large size account for the efficiency of the proposed preconditioners. Finally the potential of the parallel MCP approach is tested on a three-dimensional problem of more than 2 million unknowns and 1.2x108 nonzeros showing perfect scalability up to 256 processors.