DOCUMENTA MATHEMATICA, Extra Volume ICM III (1998), 503-512

Bjorn Engquist

Title: Wavelet Based Numerical Homogenization

In analytic homogenization, a differential equation and its solution with multiple scales are replaced by an approximating equation and its corresponding smoother solution with fewer scales. The scales related to the shortest wavelengths are eliminated. We shall start from a discretization of the original differential equation, which includes all the scales. The solution and the difference operator will be represented in a wavelet basis and the homogenized discrete operator will correspond to a particular form of an approximative projection onto the coarser scales. We shall show that this new operator inherits many of the properties of the original discrete operator, including sparseness. Some numerical examples will be presented and comparisons with the analytic homogenization process will be given. We shall also discuss direct coarse grid approximation.

1991 Mathematics Subject Classification:

Keywords and Phrases:

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