Scale dependent coarsening strategies

Solving flow and transport equations on heterogeneous multiscale porous media is a numerically demanding task. One may either reduce the complexity of the model by scaling the equations and transfer the model to large scales with a coarser resolution or improve the numerical solvers towards large scale solvers.

Ensemble averaging, volume averaging and two scale expansions are scaling methods which have been developed and successfully applied to scale up flow and transport processes in heterogeneous media. Tremendous work has been devoted to give explicit results for effective or larger scale parameters making use of perturbation theory approaches. Numerical multiscale schemes have been developedwhich step beyond perturbation theory approximations. Limitations of existing approaches are often not handled carefully which may lead to unphysical averaging effects like for example overestimated mxing in solute transport or wrong hydraulich heads in pumping tests.

In this talk, a scaling method will be introduced which is coarsening the equations in infinitesimal small coarsening steps and averaging stepwise subscale effects (Coarse Graining). Imposing the requirement of flux conservation across scales, a differential equation for the scale dependence of effective parameters in flow and transport is derived. These type of equations are well known in theoretical physics where they are called renormalisation group equations. Knowing the scale dependence of effective parameters, transient and preasymptotic effects of flow and transport through hetereogeneous media can be described more realistically avoiding unphysical averaging effects. I will present results for scale dependent effective parameters for various flow and transport problems.

The results might be also important for efficient large scale solvers like multigrid solvers. Multigrid solvers are based on restriction and interpolation schemes between coarser and finer grids and the construction of coarse grid operators is central. Knapek hypothesized that coarse grid operators in multigrids should be physically meaningful (SIAM J. Sci. Comput ,1996/) /in order to achieve good mathematical accuracy and numerical convergence rates. We tested this hypothesis for the ellliptic equation with oszillating parameters.