Hyperinterpolation for spectral wave propagation models in three dimensions

In this review article, we describe some advances in applications of the hyperinterpolation operator introduced by Sloan about two decades ago (J Approx Theory 83:238–254, 1995). In particular, our focus is on reviewing the application of the scalar and vector-valued hyperinterpolation approximations for developing, analyzing and implementing fully-discrete high-order algorithms. Such approximations facilitate efficient simulation of scattering of acoustic, electromagnetic and elastic waves, exterior to connected and disconnected bounded three dimensional domains. The main contributions of this article are: (1) a unified (acoustic, electromagnetic, and elastic) approach for the three important classes of waves; (2) theoretical and numerical comparisons of the hyperinterpolation approximations in these three applications; and (3) new results for a class of unbounded heterogeneous media.