Zeroth and first-order homogenized approximations to nonlinear diffusion through block inclusions by an analytical approach

Abstract

Approximate solutions to nonlinear diffusion systems are useful for many applications in computational science. When the heterogeneous nonlinear diffusion coefficient has high contrast values, an average solution given by upscaling the diffusion coefficient provides the average behavior of the fine-scale solution, which sometimes is infeasible to compute. This is also related to a problem that occurs during numerical simulations when it is necessary to coarsen meshes and an upscale coefficient is needed in order to build the data from the fine mesh to the coarse mesh. In this paper, we present a portable and computationally attractive procedure for obtaining not only the upscaled coefficient and the zeroth-order approximation of nonlinear diffusion systems, but also the first-order approximation which captures fine-scale features of the solution. These are possible by considering a correction to an approximate solution to the well known periodic cell-problem, obtained by a two-scale asymptotic expansion of the respective nonlinear diffusion equation. The correction allows one to obtain analytically the upscale diffusion coefficient, when the heterogeneous coefficient is periodic and rapidly oscillating describing inclusions in a main matrix. The approximate solutions provide a set of analytical basis functions used to construct the first-order approximation and also an estimate for the upper bound error implied in using the upscaled approximations. We demonstrate agreement with theoretical and published numerical results for the upscale coefficient, when heterogeneous coefficients are described by step-functions, as well as convergence properties of the approximations, corroborating with classical results from homogenization theory. Even though the results can be generalized, the emphasis is for conductivity functions of the form K ( x , u ( x ) ) = K s ( x ) k r ( u ( x ) ) , widely used for simulating flows in reservoirs.