Hybrid domain decomposition algorithms for compressible and almost incompressible elasticity

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 are uniformly bounded, which are similar to those previously developed for scalar elliptic problems and domain decomposition methods of iterative substructuring type, i.e. methods based on nonoverlapping 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 of 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 with that of the algorithm previously considered. In addition, in spite of using overlapping subdomains to define the local components of the preconditioner, values of the residual and the approximate solution need only to be retained on the interface between the subdomains 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.