<p>We consider a spectral Galerkin method for the elliptic equations with both local and nonlocal Laplacians. The presence of the nonlocal operators in the model poses significant difficulties in discretization methods and their implementation and numerical analysis. To tackle these challenges, we present a spectral Galerkin method that uses the global basis and can naturally accommodate nonlocal operators. We first prove sharp regularity estimates of solutions in weighted Sobolev spaces. Then we obtain optimal convergence orders of the spectral Galerkin methods. We also extend our method to the stochastic fractional elliptic equations with the fractional Gaussian noise. Numerical results are presented to verify the theoretical predictions.</p>