This project is aimed at performing a systematic analysis, providing a real breakthough, of the combined effect of wave propagation and numerical discretizations, in order to help in the development of efficient numerical methods mimicking the qualitative properties of continuous waves. This is an important issue for its many applications: irrigation channels, flexible multi-structures, aeronautic optimal design, acoustic noise reduction, electromagnetism, water waves, nonlinear optics, nanomechanics, etc. The superposition of the present state of the art in Partial Differential Equations (PDE) and Numerical Analysis is insufficient to understand the spurious high frequency numerical solutions that the interaction of wave propagation and numerical discretizations generates. There are some fundamental questions, as, for instance, dispersive properties, unique continuation, control and inverse problems, which are by now well understood in the context of PDE through the celebrated Strichartz and Carleman inequalities, but which are unsolved and badly understood for numerical approximation schemes. The aim of this project is to systematically address some of these issues, developing new analytical and numerical tools, which require new significant developments, much beyond the frontiers of classical numerical analysis, to incorporate ideas and tools from Microlocal and Harmonic Analysis. The research to be developed in this project will provide new analytical tools and numerical schemes. Simultaneously, it will contribute to significant progress in some applied fields in which the issues under consideration play a key role. In parallel with the analytical and numerical analysis of these problems, a mathematical simulation platform will be set to perform computer simulations and explore and visualize some of the most relevant and complex phenomena.