In this talk we will address numerical methods for two stochastic elliptic models, where the coefficients are perturbed by colored noise or white noise. An overview of the development of numerical methods will be given for the first model. We will focus on a stochastic Galerkin finite element method for the second model. Such a model is unbiased in that the expectation of the solution solves the same equation with statistically averaged coefficients. The developed numerical algorithms are based on finite element discretization in the physical space, and Wiener chaos expansion in the probability space. Since in many practically important examples solutions of the stochastic elliptic SPDEs have infinite variance, we investigate the convergence of our algorithms in appropriately weighted Wiener chaos spaces. The convergence is studied both theoretically and numerically. We also provide a comparison of the two aforementioned stochastic elliptic models through a numerical study.