A MULTIRESOLUTION STRATEGY FOR REDUCTION OF ELLIPTIC PDES AND EIGENVALUE PROBLEMS

In many practical problems coefficients of PDEs are changing across ma ny spatial or temporal scales, whereas we might be interested in the b ehavior of the solution only on some relatively coarse scale. We appro ach the problem of capturing the influence of fine scales on the behav ior of the solution on a coarse scale using the multiresolution strate gy. Considering adjacent scales of a multiresolution analysis, we expl icitly eliminate variables associated with the finer scale, which leav es us with a coarse-scale equation. We use the term reduction to desig nate a recursive application of this procedure over a finite number of scales. We present a multiresolution strategy for reduction of self-a djoint, strictly elliptic operators in one and two dimensions. It is k nown that 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 wavelet. We prove that the reduction proce dure preserves the rate of decay over any finite number of scales and therefore results in sparse matrices for computational purposes. Furth ermore, the reduction procedure approximately preserves small eigenval ues of self-adjoint, strictly elliptic operators. We also introduce a modified reduction procedure which preserves the small eigenvalues wit h greater accuracy than the standard reduction procedure and obtain es timates for the perturbation of those eigenvalues. Finally, we discuss potential extensions of the reduction procedure to parabolic and hype rbolic problems. (C) 1998 Academic Press.