N-dimensional error control multiresolution algorithms for the point value discretization

Abstract

In this paper we present N-dimensional multiresolution algorithms with and without error control strategies in the point values setting. This work complements the case of cell averages (Ruiz and Trillo, 2017). It can be somehow considered a generalization to N dimensions of several works, see Harten (1993), Harten (1996), Claypoole et al. (2003) [12], Amat et al. (2002). Apart from presenting the N-dimensional algorithms, we focus mainly on three concerns. Firstly, on deriving the corresponding results proving the stability and giving explicit error bounds. Secondly on explaining in some detail how to carry out the programming, and finally we include some interesting numerical experiments in various dimensions. In particular, we present a different application for each dimension up to N=5. They have been chosen to let clear the wide range of possible applications of these algorithms. In 1D we present an experiment where it is crucial to gain in computational time. In 2D, we deal with the compression and reconstruction of contour maps representing the topography of a certain terrain such as a seabed. In 3D, a scalar physical property, namely the salinity, is measured in a three dimensional mesh and then kept to be reconstructed later in time and compared with new measurements. In 4D, we deal with a temperature field in a cylindrical pipe in a heat exchanger. This field occupies a lot of memory inside a finite elements calculation, and a possibility to deal with this fact is to compress this field or to use the information given by the multiresolution coefficients to save computational time by refining the mesh just in the necessary zones. In 5D we work with a splitted sinusoidal function, which exemplifies a similar situation to the 2D contour map, but in a higher dimension.