Least-squares methods for PDEs: A fair and balanced perspective

Sunday, October 31, 2010 - 3:00pm - 4:30pm
Keller 3-180
Pavel Bochev (Sandia National Laboratories)
In this lecture I will present an unconventional perspective on least-squares finite element methods, which connects them to compatible methods and shows that least-squares methods can enjoy the same conservation properties as their mixed Galerkin cousins.

To a casual observer, compatible (or mimetic) methods and least
squares principles for PDEs couldn't be further apart. Mimetic
methods inherit key conservation properties of the PDE, can be
related to a naturally occurring optimization problem, and
require specially selected, dispersed degrees of freedom. The
conventional wisdom about least squares is that they rely on
artificial energy principles, are only approximately
conservative, but can work with standard C0 nodal (or collocated) degrees of freedom. The latter is considered to be among the chief reasons to use least squares methods.

This lecture demonstrates that exactly the opposite is true
about least-squares methods. First, I will argue that nodal
elements, while admissible in least squares, do not allow them
to realize their full potential, should be avoided and are,
perhaps, the least important reason to use least squares!
Second, I will show that for an important class of problems
least squares and compatible methods are close relatives that
share a common ancestor, and in some circumstances compute
identical answers. The price paid for gaining favorable
conservation properties is that one has to give up what is
arguably the least important advantage attributed to least
squares methods: one can no longer use C0 nodal elements for all variables.

If time permits I will explore two other unconventional uses of least-squares ideas which result in numerical schemes with attractive computational properties: a least-squares mesh-tying method that passes patch tests of arbitrary orders, and a locally conservative discontinuous velocity least-squares method for incompressible flows. The material in this talk is drawn from collaborative works with M. Gunzburger (FSU), M Hyman (Tulane), L. Olson (UIUC) and J. Lai (UIUC).

Sandia National Laboratories is a
multi-program laboratory operated by Sandia Corporation, a
wholly owned subsidiary of Lockheed Martin company, for the
U.S. Department of Energy's National Nuclear Security
Administration under contract DE-AC04-94AL85000.
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