Analysis of an upscaling method based on conservation of dissipation

In this paper, a scaling method based on conservation of dissipation and use of periodic boundary conditions is presented. We prove that the method leads to a symmetric positive definite tensor. We also show that the method is identical to the method of Durlovsky and Chung, therefore an important property of the latter method is proved. Some other existing methods are also discussed in terms of their boundary conditions. For this purpose, the concepts of basis and class of boundary conditions for scaling methods are introduced.