Multiresolution Reduction of Elliptic PDE's and Eigenvalue Problems

Nicholas Coult

Department of Applied Mathematics Campus Box 526 University of Colorado Boulder, CO 80309


Abstract

Coefficients of PDE's are often changing across many spatial or temporal scales, whereas we might be interested in the behavior of the solution only on some relatively coarse scale. The multiresolution strategy for reduction explicitly solves for the fine-scale behavior in terms of the coarse scale, leaving us with a coarse-scale equation. We present a multiresolution strategy for reduction of elliptic operators, and demonstrate the importance of using high order wavelets in the reduction procedure. It is known tha the non-standard form for a wide class of operators has fast off-diagonal decay and the rate of decay is controlled by the number of vanishing moments of the wavelets; we prove that the reduction procedure preserves the rate of decay on all scales, and, therefore, results in sparse matrices for computational purposes. Furthermore, the reduction procedure approximately preserves small eigenvalues of elliptic operators. We introduce a modified reduction procedure which preserves the small eigenvalues with greater accuracy than the standard reduction procedure and obtain estimates for the perturbation of those eigenvalues. Finally, we discuss extension of the reduction procedure to hyperbolic problems and to elliptic operators in higher spatial dimensions.