Wavelet-based Homogenization of Di erential Operators

The exact solutions of PDEs with ne-scale co-eecients and data contain details that make computations expensive. When only the coarse-scale approximations of the solutions are needed simpler equations can be derived in a process called homogenization. Classical homogenization techniques are based on the analytic expansion of the exact solution. There are also limitations, such as periodic-ity of the ne-scale. In the wavelet hierarchy of embedded spaces, the notion of ne-and coarse-scale features are rigorous, and projection onto these spaces is a fast operation. A discrete operator is naturally split by the fast wavelet transform into ne-and coarse-scale components and the homogenized operator is obtained as a Schur complement. The matrix manipulation is eeciently encoded in Matlab. The structure of discrete divergence-form el-liptic operators is invariant under wavelet ho-mogenization. The point-wise multiplication operator is a matrix well approximated by a band-diagonal matrix. The homogenized operator is approximated by sparse matrices, reducing the computational complexity. For periodic ne-scale coeecients, wavelet homoge-nization produces a discretization of the eeec-tive equations. The wavelet technique is exi-ble since it can handle localized ne scale variations and multiple scales.