  Stochastic Galerkin finite element discretizations of partial differential
equations with coefficients characterized by arbitrary distributions lead, in
general, to fully block dense linear systems. We propose two novel strategies
for constructing preconditioners for these systems to be used with Krylov
subspace iterative solvers. In particular, we present a variation on of the
hierarchical Schur complement preconditioner, developed recently by the
authors, and an adaptation of the symmetric block Gauss-Seidel method. Both
preconditioners take advantage of the hierarchical structure of global
stochastic Galerkin matrices, and also, when applicable, of the decay of the
norms of the stiffness matrices obtained from the polynomial chaos expansion of
the coefficients. This decay allows to truncate the matrix-vector
multiplications in the action of the preconditioners. Also, throughout the
global matrix hierarchy, we approximate solves with certain submatrices by the
associated diagonal block solves. The preconditioners thus require only a
limited number of stiffness matrices obtained from the polynomial chaos
expansion of the coefficients, and a preconditioner for the diagonal blocks of
the global matrix. The performance is illustrated by numerical experiments.
