William Symes, Rice University
KAUST-IAMCS Workshop on Modeling and Simulation of Wave Propagation and Applications 2012
May 8-9, 2012
King Abdullah University of Science and Technology (KAUST)
Thuwal, Kingdom of Saudi Arabia
Upscaling of Wave Propagation and Harmonic Maps
Harmonic maps arise naturally in the theory of homoginization. Recently, Owhadi and Zhang suggested a Galerkin discretization of the 2D acoustic wave equation, using the pull-back by a harmonic map of a standard low-order conforming basis. Unlike standard Galerkin discretization, the Owhadi-Zhang approximation converges at a positive rate not depending on any smoothness constraints on the coefficients. The rate is however suboptimal, compared with that achieved by the standard Galerkin approach with piecewise polynomial conforming basis for smooth coefficient problems. In their recent PhD work, Binford and Wang showed how to modify the Owhadi-Zhang algorithm to recover optimal convergence in the static and dynamic cases respectively. Wang also showed that mass lumping does not affect the order of convergence. The Owhadi-Zhang construction via harmonic pull-back seems limited to scalar problems, and in particular does not extend in any obvious way to linear elasticity. Composition of elastodynamic wavefields with the obvious harmonic map does not seem to induce sufficient smoothness to imply optimal order convergence. A different approach to establishing convergence seems to be required; I will finish with a few thoughts about one such alternative.