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
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
) for all variables and optimal
convergence for the scalar variable.
However, showing optimal
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
accuracy.
In this talk, we show that optimal 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
-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
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.
This seminar is easily accessible to persons with disabilities.
For more information or for assistance, please contact the Mathematics
Department at 743-3500.