### Lazarov R. Minev P. Srinivasan S.

## Numerical upscaling of transient flows in heterogeneous porous media by direction-splitting technique

### Reporter: Lazarov R.

We present and discuss two methods for numerical upscaling (see, e.g. [1]) 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 multi-scale finite volume and finite element methods in the context of the direction splitting approach [3]. The second method devises the approximation from the Schur complement corresponding to the interface unknowns of the coarse grid, by applying a proper projection operator to a certain trace space. 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 (e.g. [2]). 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 tested numerically on benchmark coefficient data available in the literature. The paper is based on our report [4].

The project has been partially supported by grants USA NSF DMS-1016525.

REFERENCES

1. Efendiev Y., Hou T., Multiscale Finite Element Methods: Theory and Applications, V. 4 of Surveys and Tutorials in the Applied Mathematical Sciences, New York: Springer, 2009.

2. Laevsky, Yu.M., Quadratic elements in splitting methods // Sov. J. Numer. Anal. And Math. Model. 1990. V.5. P. 244—249.

3. Marchuk G.I., Splitting and Alternating Direction Methods, Handbook of Numerical Analysis, Amsterdam: North-Holland, 1990, vol. 1, P. 197—464.

4. Srinivasan S., V., Lazarov R., Minev P., Multiscale Direction-Splitting Algorithms for Parabolic
Equations with Highly Heterogeneous Coefficients, arxiv: 2000:2000, 2015.

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