The Institute for Theoretical and Engineering Science
Department of Mathematics, University of Houston

Scientific Computing Seminars

Dr. Pavel Bochev
Computational Mathematics and Algorithms
Sandia National Laboratories

On least-squares principles for the Poisson equation
and their connection to the Dirichlet and Kelvin principles,
or how to do least-squares finite elements right.

Monday, November 22$^*$
3:00 PM- 4:00 PM
Room 634 S&R1

$^*$Note: Special Date

Abstract: Historically, least-squares variational principles have been viewed solely as a way to avoid the inf-sup condition in the mixed finite element method, so as to allow the use of standard $C^0$ equal-order nodal elements for all dependent variables. And indeed, with regard to this goal, least-squares principles have consistently delivered stable and robust discretizations.

For example, least-squares finite element methods for first-order formulations of the Poisson equation are not subject to the inf-sup condition and lead to stable solutions even when all variables are approximated by equal-order, continuous finite element spaces. For such elements, one can also prove optimal convergence in the ``energy'' norm (equivalent to a norm on $H^{1}(\Omega)\times H(\Omega,\rm div)$) for all variables and optimal $L^2$ convergence for the scalar variable. However, showing optimal $L^2$ convergence for the flux has proven to be impossible without adding the redundant curl equation to the first-order system of partial differential equations. In fact, numerical evidence strongly suggests that nodal, continuous flux approximations do not posses optimal $L^2$ accuracy.

In this talk, we show that optimal $L^2$ error rates for the flux can be achieved without the curl constraint provided one uses the div-conforming family of Brezzi-Douglas-Marini or Brezzi-Douglas-Duran-Fortin elements. Then, we proceed to reveal an interesting connection between a least-squares finite element method involving $H(\Omega,\rm div)$-conforming flux approximations and mixed finite element methods based on the classical Dirichlet and Kelvin principles. We show that such least-squares finite element methods can be obtained by approximating, through an $L^2$ projection, the Hodge * operator that connects the Kelvin and Dirichlet principles. Our principal conclusion is that when implemented in this way, a least-squares finite element method combines the best computational properties of mixed finite element methods based on each of the classical principles.

$\bullet$ This seminar is easily accessible to persons with disabilities. For more information or for assistance, please contact the Mathematics Department at 743-3500.