Authors: Luca F. Pavarino, Olof B. Widlund, and Stefano Zampini


Title: BDDC preconditioners for spectral element discretizations of
almost incompressible elasticity in three dimensions

 Abstract: 
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
FETI-DP algorithms defined by the same set of primal constraints.
Numerical experiments carried out on parallel computing systems
confirm these results.