Finite Element Approximation of Partial Differential Equations

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The first area of study is the approximation properties of several types offinite element spaces defined on irregular hexahedral elements obtained bytrilinear mappings from a reference cube. Such spaces are used to approximate three-dimensional vector functions and arise naturally in many applications,including the approximation of Maxwell's equations and the use of mixed andleast squares finite element methods for second order elliptic equations. The research is to determine precisely what is needed for optimal orderapproximation and construct families of finite element spaces that have thisproperty. The second area of study is the finite element approximation bydiscontinuous Galerkin methods of convection-diffusion problems. The aim isto derive new local error estimates in order to understand which methods ofthis promising class of approximation schemes work well both fordiffusion-dominated and convection-dominated second order partial differential equations. The third area of research is to use the now well-developed theory for the approximation of the Reissner-Mindlin plate model (used to study thebending of a thin plate under external loads) as a basis for developing newapproaches to the use of finite element methods for the approximation ofelastic shells. Both the plate model and to a greater extent the shell model suffer from the problem of "locking'' when standard finite elementapproximation schemes are applied, causing poor approximations for thin plates and shells. The final area of research involves the design of effectivenumerical methods for the Einstein equations, used to numerically simulate the emission of gravitation radiation from massive astronomical events such asblack hole collisions. The approach taken will be to use simpler modelproblems with some of the same features to understand why standard numericalmethods for the Einstein equations fail and to help design methods thatovercome these problems.The mathematical modeling of physical and biological processes using partialdifferential equations has become the standard method of studying a host ofimportant scientific problems. Since it is usually not possible to solve such equations exactly, the development of reliable and efficient numericalapproximation schemes, which can be implemented on computers, makes this into a practical approach and is central to progress in many areas of science andengineering. This project studies "finite element" type approximation schemes for mathematical models of a variety of applied problems. These include flows of gases and fluids in which both convection and diffusion are present,Maxwell's equations for the modeling of the electric and magnetic fields in a body subject to an applied current, the bending of thin structures (e.g., aroof) under external loads, and Einstein's equations for the simulation of the emission of gravitation radiation from massive astronomical events such asblack hole collisions. This work is expected to lead to new and improvednumerical methods for use by scientists and engineers in applied computations.