Project Details
Description
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.
Status | Finished |
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Effective start/end date | 8/1/03 → 7/31/07 |
Funding
- National Science Foundation: $172,375.00