Displacement decomposition circulant preconditioners for almost incompressible 2D elasticity systems

Abstract

The robustness of the recently introduced circulant blockfactorization (CBF) preconditioners is studied in the case of finite element matrices arising from the discretization of the 2D Navier equations of elasticity. Conforming triangle finite elements are used for the numerical solution of the differential problem. The proposed preconditioner M _C is constructed by CBF approximation of the block-diagonal part of the stiffness matrix. In other words, we implement in our algorithm the circulant block-factorization into the framework of the displacement decomposition technique. The estimate κ(M ⁻¹_C K)=O(√N/1−v) is proved asymptotically on N, where N is the size of the discrete problem. Note, that the corresponding known estimate for the widely used incomplete factorization displacement decomposition preconditioner M _ILU is κ(M ⁻¹ _ILU K)=O(√N/1−v)

The theoretical estimate as well as the presented numerical tests show some significant advantages of this new approach for a PCG iterative solution of almost incompressible elastic problems, that is when the modified Poisson ratio v tends to the incompressible limit case v=1.

This paper was partially supported by the Bulgarian Ministry of Education, Science and Technology under grant MM 417/94.