Iterative substructuring methods for spectral element discretizations of elliptic systems I: Compressible linear elasticity

An iterative substructuring method for the system of linear elasticity in three dimensions is introduced and analyzed. The pure displacement formulation for compressible materials is discretized with the spectral element method. The resulting stiffness matrix is symmetric and positive definite. The proposed method provides a domain decomposition preconditioner constructed from local solvers for the interior of each element and for each face of the elements and a coarse, global solver related to the wire basket of the elements. As in the scalar case, the condition number of the preconditioned operator is independent of the number of spectral elements and grows as the square of the logarithm of the spectral degree.