Mathematics Colloquia and Seminars

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We consider the numerical solution of, mainly linear, partial differential equations with coefficients that vary on length scales much smaller than the computational domain. The coefficients could represent for instance rapid variations in material properties or small details in the geometry of the problem. A direct numerical simulation will in general not be accurate if the computational grid is coarser than those small length scales. The simulation would ignore the effect of small scale processes on the solution. However, using a fine enough grid can be computationally too expensive. We show how one can instead derive an ``effective'' discrete operator for the coarse grid simulation, in analogy with classical homogenization theory for the continuous problem. The structure of this effective operator is similar to the one given by direct discretization of the differential equations on the coarse grid, but it has a locally altered stencil that takes the influence of subgrid scales into account. We discuss a general procedure for deriving such operators, based on wavelet projections of the discrete fine grid operator followed by sparse approximation. We review some theoretical results related to this approach and show numerical examples.