Finite Element Approximation of Partial Differential Equations

Rutgers University New Brunswick, New Brunswick NJ

Investigators

Abstract

The first area of study is the approximation properties of several types of finite element spaces defined on irregular hexahedral elements obtained by trilinear 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 and least squares finite element methods for second order elliptic equations. The research is to determine precisely what is needed for optimal order approximation and construct families of finite element spaces that have this property. The second area of study is the finite element approximation by discontinuous Galerkin methods of convection-diffusion problems. The aim is to derive new local error estimates in order to understand which methods of this promising class of approximation schemes work well both for diffusion-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 the bending of a thin plate under external loads) as a basis for developing new approaches to the use of finite element methods for the approximation of elastic shells. Both the plate model and to a greater extent the shell model suffer from the problem of "locking'' when standard finite element approximation schemes are applied, causing poor approximations for thin plates and shells. The final area of research involves the design of effective numerical methods for the Einstein equations, used to numerically simulate the emission of gravitation radiation from massive astronomical events such as black hole collisions. The approach taken will be to use simpler model problems with some of the same features to understand why standard numerical methods for the Einstein equations fail and to help design methods that overcome these problems. The mathematical modeling of physical and biological processes using partial differential equations has become the standard method of studying a host of important scientific problems. Since it is usually not possible to solve such equations exactly, the development of reliable and efficient numerical approximation schemes, which can be implemented on computers, makes this into a practical approach and is central to progress in many areas of science and engineering. 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., a roof) under external loads, and Einstein's equations for the simulation of the emission of gravitation radiation from massive astronomical events such as black hole collisions. This work is expected to lead to new and improved numerical methods for use by scientists and engineers in applied computations.