A multiresolution strategy for numerical homogenization
The work is concerned with a multiresolution strategy for homogenization of differential equations (ordinary and partial). The authors develop an efficient numerical approach which generates the coefficients of the homogenized equation. The multiresolution analysis (MRA), a notion introduction by \textit{Y. Meyer}, is used [Rev. Mat. Iberoam. 8, No. 2, 115-133 (1991; Zbl 0753.42015)]. The equations (differential or integral) considered in this article can be written in the general form (1) \(Bx + q + \lambda = K(Ax + p)\) where \(A\), \(B\), \(K\) are operators on functions in \(L_2(0,1)\) with values in a given Hilbert space \(H\), \(p\), \(q\), \(x\) are square-integrable functions defined on \([0,1]\) with values in \(H\) and \(\lambda\) is a parameter. The MRA strategy of \(L_2(0,1)\) consists in representation of the space \(L_2(0,1)\) as a direct sum of some subspaces \(V_n\) and their orthogonal complements \(W_n\). Then, the equation (1) is discretized by applying the projection operators to operators \(A\), \(B\), \(K\), and using new notation some recursion relations are obtained. The steps of the procedure are described and convergence results a presented. After general results, a special type (matrix) Volterra integral equation is treated as an application, using a Haar basis. An algorithm for generating homogenized equations is given. Three numerical examples are discussed and the solutions are compared to other numerical methods.
scientific article; zbMATH DE number 846191