Multiscale direction-splitting algorithms for parabolic equations with highly heterogeneous coefficients

In this paper we discuss two methods for upscaling of highly heterogeneous data for parabolic problems in the context of a direction splitting time approximation. The first method is a direct application of the idea of Jenny et al. (2003) in the context of the direction splitting approach. The second method devises the approximation from the Schur complement corresponding to the interface unknowns of the coarse grid, by applying a proper L2 projection operator to it. The spatial discretization employed in this paper is based on a MAC finite volume stencil but the same approach can be implemented within a proper finite element discretization. A key feature of the present approach is that it can extend to 3D problems with very little computational overhead. The properties of the resulting approximations are demonstrated numerically on some benchmark coefficient data available in the literature.