  We derive computable error estimates for finite element approximations of
linear elliptic partial differential equations (PDE) with rough stochastic
coefficients. In this setting, the exact solutions contain high frequency
content that standard a posteriori error estimates fail to capture. We propose
goal-oriented estimates, based on local error indicators, for the pathwise
Galerkin and expected quadrature errors committed in standard, continuous,
piecewise linear finite element approximations. Derived using easily validated
assumptions, these novel estimates can be computed at a relatively low cost and
have applications to subsurface flow problems in geophysics where the
conductivities are assumed to have lognormal distributions with low regularity.
Our theory is supported by numerical experiments on test problems in one and
two dimensions.
