Abstract: Spectral element methods are considered for symmetric elliptic systems of second-order partial differential equations, such as the linear elasticity and the Stokes systems in three dimensions. The resulting discrete problems can be positive definite, as in the case of compressible elasticity in pure displacement form, or saddle point problems, as in the case of almost incompressible elasticity in mixed form and Stokes equations. Iterative substructuring algorithms are developed for both cases. They are domain decomposition preconditioners 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. In the positive definite case, the condition number of the resulting preconditioned operator is independent of the number of spectral elements and grows at most in proportion to the square of the logarithm of the spectral degree. For saddle point problems, there is an additional factor in the estimate of the condition number, namely, the inverse of the discrete inf-sup constant of the problem.