BDDC Preconditioners for Spectral Element Discretizations of Almost Incompressible Elasticity in Three Dimensions

Balancing domain decomposition by constraints (BDDC) algorithms are constructed and analyzed for the system of almost incompressible elasticity discretized with Gauss–Lobatto–Legendre spectral elements in three dimensions. Initially mixed spectral elements are employed to discretize the almost incompressible elasticity system, but a positive definite reformulation is obtained by eliminating all pressure degrees of freedom interior to each subdomain into which the spectral elements have been grouped. Appropriate sets of primal constraints can be associated with the subdomain vertices, edges, and faces so that the resulting BDDC methods have a fast convergence rate independent of the almost incompressibility of the material. In particular, the condition number of the BDDC preconditioned operator is shown to depend only weakly on the polynomial degree $n$, the ratio $H/h$ of subdomain and element diameters, and the inverse of the inf-sup constants of the subdomains and the underlying mixed formulation, while being scalable, i.e., independent of the number of subdomains and robust, i.e., independent of the Poisson ratio and Young's modulus of the material considered. These results also apply to the related dual-primal finite element tearing and interconnect (FETI-DP) algorithms defined by the same set of primal constraints. Numerical experiments, carried out on parallel computing systems, confirm these results.