Hybrid Domain Decomposition Algorithms for Compressible and Almost Incompressible Elasticity Tr2008-919

Overlapping Schwarz methods are considered for mixed finite element approximations of linear elasticity, with discontinuous pressure spaces, as well as for compressible elasticity approximated by standard conforming finite elements. The coarse components of the preconditioners are based on spaces, with a number of degrees of freedom per subdomain which is uniformly bounded, and which are similar to those previously developed for scalar elliptic problems and domain decomposition methods of iterative substructuring type, i.e., methods based on non-overlapping decompositions of the domain. The local components of the new preconditioners are based on solvers on a set of overlapping subdomains. In the current study, the dimension of the coarse spaces is smaller than in recently developed algorithms; in the compressible case all independent face degrees of freedom have been eliminated while in the almost incompressible case five out six are not needed. In many cases, this will result in a reduction of the dimension of the coarse space by about one half compared to that of the algorithm previously considered. In spite of using overlapping subdomains to define the local components of the preconditioner, only values on the interface between the subdomains need to be retained in the iteration of the new hybrid Schwarz algorithm. The use of discontinuous pressures makes it possible to work exclusively with symmetric, positive definite problems and the standard preconditioned conjugate gradient method. Bounds are established for the condition number of the preconditioned operators. The bound for the almost incompressible case grows in proportion to the square of the logarithm of the number of degrees of freedom of individual subdomains and the third power of the relative overlap between the overlapping subdomains, and it is independent of the Poisson ratio as well as jumps in the Lamé parameters across the interface between the subdomains. Numerical results illustrate the findings.