@TechReport{ it:2002-001,
author = {S{\^o}nia M. Gomes and Bertil Gustafsson},
title = {Combining Wavelets with Finite Differences: Consistency
Analysis},
institution = {Department of Information Technology, Uppsala University},
department = {Division of Scientific Computing},
year = {2002},
number = {2002-001},
month = jan,
abstract = {The method is an adaptive finite difference strategy for
numerical solution of evolution partial differential
equations. The principle is to represent the solution only
through those point values indicated by the significant
wavelet coefficients. Typically, few points are found in
each time step, the grid being coarse in smooth regions,
and refined close to irregularities. At each point, the
spatial derivatives are discretized by uniform finite
differences, using step size proportional to the point
local scale. Eventually, required neighboring stencils are
not present in the grid. In such case, the correspondig
point values are approximated from coarser scales by using
reconstruction operators defined by means of interpolating
subdivision scheme. Our purpose in this paper is to analyse
a generalization of the concept of truncation error, which
is the familiar basis of the analysis of difference
schemes. For this consistency analysis, we show that the
adaptive finite difference scheme can also be formulated in
terms of a collocation scheme for an adapted wavelet
expansion of the solution. For this purpose, we first prove
some results concerning the local behavior of the
reconstruction operators, which stand for appropiate
cone-like grids. }
}