Numerical Analysis and Simulation of Geometric Wave Equations

Numerical simulations have grown to become a significant component of almost all scientific disciplines, playing the role of physical experiment in many circumstances. Yet for many mathematical models and in particular in the new field of gravitational wa ve astronomy, current algorithms are unsatisfactory, probably because of a lack of understanding of the relationship between continuum models and discrete models. We address in this project the problem of designing good discretizations of geometric wave e quations, including scalar waves, (classical) Yang-Mills equations and Einstein's equations of general relativity. With various degrees of non-linearity and difficulty such equations nevertheless have in common a rich algebraic structure, inherited from d ifferential geometry and including invariance under certain groups of transformations. We will develop tools enabling the numerical analysis of discretizations of such equations, drawing on the experience of finite element discretizations of partial diffe rential equations and structure preserving algorithms for ordinary differential equations. While finite element theory has been mainly developed in the framework of Sobolev spaces we aim to adapt some of the more elaborate methods of analysis developed fo r non-linear equations in the calculus of variations. Mimicking the algebraic structure of the continuum model in the construction of algorithms is a major trend in contemporary numerical analysis which has yet to realize its potential for partial differe ntial equations where the interplay with topological properties becomes crucial. Geometric wave equations provide a field of choice to bridge this gap on physically relevant systems.