  The paper considers a class of parametric elliptic partial differential
equations (PDEs), where the coefficients and the right-hand side function
depend on infinitely many (uncertain) parameters. We introduce a two-level a
posteriori estimator to control the energy error in multilevel stochastic
Galerkin approximations for this class of PDE problems. We prove that the
two-level estimator always provides a lower bound for the unknown approximation
error, while the upper bound is equivalent to a saturation assumption. We
propose and empirically compare three adaptive algorithms, where the structure
of the estimator is exploited to perform spatial refinement as well as
parametric enrichment. The paper also discusses implementation aspects of
computing multilevel stochastic Galerkin approximations.
