Numerical procedures are discussed for constructing averaged coefficients for elliptic differential operators. These procedures are intended for problems where the coefficients vary on a scale finer than can be resolved by a reasonable computational grid. Numerical methods for calculating locally averaged coefficients using mixed and Galerkin finite elements are presented. These methods involve solving local elliptic problems either to determine a pseudo-coefficient or as part of the overall solution procedure. The local problems are independent and can be solved in parallel. The procedures are formulated and numerical results demonstrating their performance are presented.