Iterative substructuring methods for spectral element discretizations of elliptic systems. II: Mixed methods for linear elasticity and Stokes flow

Iterative substructuring methods are introduced and analyzed for saddle poi nt problems with a penalty term. Two examples of saddle point problems are considered: The mixed formulation of the linear elasticity system and the g eneralized Stokes system in three dimensions. These problems are discretize d with spectral element methods. The resulting stiffness matrices are symme tric and indefinite. The interior unknowns of each element are first implic itly eliminated by using exact local solvers. The resulting saddle point Sc hur complement is solved with a Krylov space method with block precondition ers. The velocity block can be approximated by a domain decomposition metho d, e.g., of wire basket type, which is constructed from a local solver for each face of the elements, and a coarse solver related to the wire basket o f the elements. The condition number of the preconditioned operator is inde pendent of the number of spectral elements and is bounded from above by the product of the square of the logarithm of the spectral degree and the inve rse of the discrete inf-sup constant of the problem.