First-Order Systems Least Squares:
A methodology for solving systems of PDEs
Thomas A. Manteuffel
University of Colorado at Boulder
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 $H^1$ product
norm, then optimal multigrid performance is guaranteed.
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. The following talks will
present results on specific applications.