First-Order Systems Least Squares:
A Methodology for Solving Systems of PDEs
by
Tom Manteuffel
University of Colorado at Boulder
Coauthors: Steve McCormick

The process of modeling a physical system involves creating a mathematical model, forming a discrete approximation, and solving the resulting linear or nonlinear system. The mathematical model may take many forms. The particular form chosen may greatly influence the ease and accuracy with which it may be discretized as well as the properties of the resulting linear or nonlinear system. If a model is chosen incorrectly it may yield linear systems with undesirable properties such as nonsymmetry or indefiniteness. On the other hand, if the model is designed with the discretization process and numerical solution in mind, it may be possible to avoid these undesirable properties.

This talk will discuss a methodology for solving systems of partial differential equations. The methodology involves expanding the original system as a system of first-order equations by introducing new variables, adding extra constraints, and constructing a least-squares functional. If it can be shown that the least-squares functional is V-elliptic in a convenient norm, then Lax-Milgram theory guarantees that the minimization problem associated with the functional has a unique solution. In other words, the minimization problem is well posed in the V-norm.

In this context, discrete approximations to the minimum of the functional can be easily addressed through restricting the minimization to a finite element space in V. Cea's Lemma and interpolation theory now yield discretization error estimates.

Any basis for the finite element space leads to a symmetric positive definite linear system for the solution of the discrete minimization problem. If the basis has local support, then the condition of the system will be O(h^-2) because the functional involves only first-order differential operators. Moreover, if the functional can be shown to be V-elliptic in an H¹ product norm, then optimal multigrid performance is guaranteed.

An additional advantage of this approach is that it yeilds very accurate local a posterori error estimates which can be used to construct a multilevel local adaptive refinement procedure.

This methodology has been applied to a variety of applications including advection-diffusion equations, transport of neutral particles, Helmholtz equations, Stokes and Navier Stokes equations and linear elasticity. This talk will attempt to describe the basic methodology, demonstrate its use on several important applications and dicuss numerical results for linear elasticity.

http://amath-www.colorado.edu/appm/faculty/tmanteuf/

Date received: July 7, 1999

Copyright © 1999 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # cadk-15.