Numerical Solution of Partial Differential Equations and Applications

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The investigator will devise, improve, and analyze methods for the numerical simulation of complex physical phenomena modeled by partial differential equations, emphasizing three main areas: mixed methods for elasticity equations, discontinuous Galerkin methods, and computational general relativity. For elasticity, the investigator will build on a recent breakthrough that enabled the construction of the first stable mixed finite element methods for the displacement-stress formulations with polynomial trial functions, and also work towards the development of simpler nonconforming mixed finite element methods and extensions to three dimensional elasticity problems. Concerning discontinuous Galerkin methods--finite element methods in which the approximating piecewise polynomial functions are discontinuous, with modifications incorporated into the variational formulation to achieve consistency--the investigator will work with his collaborators to build on recent work on the unification and classification of such methods to develop a unified approach to the analysis of and discrimination among a wide class of discontinuous Galerkin methods for elliptic equations. Application will also be made to the numerical simulation of elastic plates incorporating shear. The third and largest effort concerns the numerical solution of Einstein's field equations relating mass and the curvature of space-time. The emphasis here will be on understanding the fundamental properties of the Einstein equations most relevant to their numerical solution and the basic difficulties that have beset previous attempts at numerical simulations of them. The work will be guided by the goal of simulating the coalescence of inspiralling pairs of black holes and the resulting emission of gravitational radiation, which is a problem of fundamental importance to gravitational physics and also because such simulations will be essential to realization of a new generation of observatories based on gravitational wave detectors.Computer simulation is a key tool for the design and testing of complex engineering structures. In recent decades computer simulation has also joined experiment and theory as one of the main paradigms of scientific investigation. In both areas, many of the most complex systems are first modeled by systems of partial differential equations--in which the language of calculus is used to express the variations of the relevant physical quantities in space and time--and then these systems of differential equations must be approximated by numerical algorithms, which harness the power of modern computers to perform billions of arithmetical operations a seconds to extract the solutions to the equations to the required degree of accuracy. In recent decades the principles for the design and validation of such algorithms have been developed for many of the basic systems of differential equations encountered in science in technology, but many more complex systems have so far resisted effective computation, and that is the thrust of this research. A particular emphasis will be on numerical algorithms for accurate determination of the stresses internal to elastic structures, which is essential to building safe and economic engineering structures. A second emphasis will be on developing methods to simulate Einstein's equations of general relativity, especially for predicting the output of gravitational radiation--minute ripples that propagate on the curved surface of space-time--from massive cosmological events such as black hole collisions. Computer codes capable of making such predictions are needed to realize the effectiveness of a new type of observatory based on gravitational radiation currently being constructed, which will provide mankind with its first window on the dark matter that makes up 90% of the universe.