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Computational Science, Engineering & Technology Series
ISSN 17593158 CSETS: 26
DEVELOPMENTS AND APPLICATIONS IN ENGINEERING COMPUTATIONAL TECHNOLOGY Edited by: B.H.V. Topping, J.M. Adam, F.J. Pallarés, R. Bru and M.L. Romero
Chapter 15
Numerical Methods for Saddle Point Problems arising in Geomechanics L. Bergamaschi
Department of Mathematical Methods and Models for Scientific Applications, University of Padua, Italy L. Bergamaschi, "Numerical Methods for Saddle Point Problems arising in Geomechanics", in B.H.V. Topping, J.M. Adam, F.J. Pallarés, R. Bru and M.L. Romero, (Editors), "Developments and Applications in Engineering Computational Technology", SaxeCoburg Publications, Stirlingshire, UK, Chapter 15, pp 337361, 2010. doi:10.4203/csets.26.15
Keywords: preconditioning, saddle point, Krylov subspace methods, coupled consolidation.
Summary
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 illconditioning, 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 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 P_{A} of the (1,1) block A and P_{B} of the resulting Schur complement matrix S = B P_{A}^{1} B\T + C. Bergamaschi et al. [1,2] have developed an MCP (mixed constraint preconditioner) which couples two explicitimplicit 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 P_{A} and P_{B}. It is found that the eigenvalues, most of which are complex ones, are well clustered around unity, provided that eigenvalues of P_{A}^{1}A and P_{S}^{1}S 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 threedimensional problem of more than 2 million unknowns and 1.2x10^{8} nonzeros showing perfect scalability up to 256 processors. References
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