Finite Element Approximation of Partial Differential Equations

  • Falk, Richard (PI)

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.

StatusFinished
Effective start/end date8/1/037/31/07

Funding

  • National Science Foundation: $172,375.00

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