# 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

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

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.

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

*C*^{0}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

*C*^{0}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.

MSC Code:

93E24

Keywords: