Generalized multiresolution analysis on unstructured grids

Efficiency of high-order essentially non-oscillatory (ENO) approximations o f conservation laws can be drastically improved if ideas of multiresolution analysis are taken into account. These methods of data compression not onl y reduce the necessary amount of discrete data but can also serve as tools in detecting local low-dimensional features in the numerical solution. We d escribe the mathematical background of the generalized multiresolution anal ysis as developed by Abgrall and Harten in [14], [15] and [3]. We were able to ultimately reduce the functional analytic background to matrix-vector o perations of linear algebra. We consider the example of interpolation on th e line as well as the important case of multiresolution analysis of cell av erage data which is used in finite volume approximations. In contrast to Ab grall and Harten, we develop a robust agglomeration procedure and recovery algorithms based on least-squeare polynomials. The efficiency of our algori thms is documented by means of several examples.