An analytical coarse grid operator applied to a multiscale multigrid method

Equations describing flow through heterogeneous porous media are often difficult to solve by direct numerical simulation given their expensive computational requirement. In this paper, numerical convergence is achieved for solving the flow equation with high contrast variable coefficients using two low computational methods that incorporate analytical approximations into the geometric multigrid. The methods are inheritably fast and easily implementable with coefficients describing general non-periodic media. The first method, named here as Analytical Coarse Operator (ACO), defines the operator at each coarse-scale of the V-cycle using an analytical approximation of the upscale tensor, it works in the same way as iterative homogenization. Its efficiency and reliability are measured by comparing with similar algorithms using arithmetic and harmonic averages of the coefficient, since they are widely used in the literature. The second method, called Analytical Prolongation Operator (APO), may be termed as aggressive coarsening, because it obtains the solution by skipping few levels in the V-cycle. It is an extension of the first approach where a multiscale prolongation operator is defined using an analytical approximation of the solution of the cell-problem from homogenization theory. The efficiency and reliability of this method are measured by comparing the number of V-cycles with the full numerical implementation. Various test cases demonstrated that convergence can be achieved using typical implementation procedures of the geometric multigrid method, therefore it highlights the low cost and easy implementation features of the methods. The examples include media having separable and non separable scales, periodic or not. In the case of non periodic medium, the problem of determining coarse grids that are representative element volume (REV) is also addressed. The medium from the SPE10 benchmark project is an application of the method for non separable scale.