Workshop on

numerical methods for multiscale problems

 

     
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  Todd Arbogast: Two-scale, locally conservative variational multiscale methods for elliptic problem

We present a two-scale theoretical framework for approximating the solution of a second order elliptic problem. The elliptic coefficient is assumed to vary on a scale that can be resolved on a fine numerical grid, but that limits on computational power require that computations be performed on a coarse grid. We consider the elliptic problem in mixed variational form, and base our scale expansion on local mass conservation over a coarse grid. The coarse grid is used to define a direct sum decomposition of the solution space into coarse and ``subgrid'' subspaces satisfying the usual divergence constraints and the property that the subgrid space is locally supported over the coarse mesh. We then explicitly decompose the variational problem into coarse and subgrid scale problems. The subgrid problem gives a well defined operator taking the coarse space to the subgrid which is localized in space, and it is used to upscale, that is, to remove the subgrid from the coarse scale problem. Using standard mixed finite element spaces, two-scale mixed spaces are defined. A mixed approximation is defined, which can be viewed as a type of variational multiscale method or a residual free bubble technique. A numerical Greens function approach is used to make the approximation to the subgrid operator efficient to compute. A mixed method $\pi$-operator is defined for the two-scale approximation spaces and used to show optimal order error estimates. Moreover, both the coarse and fine-scale problems remain locally conservative. Numerical examples representing flow in a porous medium are presented to illustrate the effectiveness and applicability of the method. We also consider the use of the approximation as a preconditioner for a direct approximation of the full problem.
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