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{\large Chris Siefert \\
{\bf An Algebraic Multigrid Method for Least-Squares Finite Element Systems}}

Sandia National Laboratories \\ P O Box 5800 \\ MS 0378 \\ Albuquerque \\ NM 87185-0378
\\
{\tt csiefer@sandia.gov}\\
Pavel Bochev\\
{\em Sandia National Laboratories}\\
Kara Peterson\\
{\em Sandia National Laboratories}\end{center}

Least-squares finite element methods are an attractive class of methods
for the numerical solution of certain partial differential equations,
specifically div-curl systems. They are motivated by the desire to
recover, in more general settings, the advantageous features of
Rayleigh-Ritz methods such as the avoidance of discrete compatibility
conditions and the production of symmetric and positive definite discrete
systems. The methods are based on the minimization of convex functionals
that are constructed from equation residuals.

This work deals with formulation of a class of compatible
least-squares principles for a model div-curl equation and the
efficient multigrid solution of the resulting algebraic
systems. Specifically, we consider a discrete least-squares principle
\begin{equation}\label{eq1}
\min_{u^h\in C^h_0}
\| \nabla \times u^h - g \|^2_0 + \| \nabla_h^* \cdot u^h - f\|^2_0
\end{equation}
where $C^h_0$ is a curl-conforming space constrained by the
homogeneous boundary condition $u\times {\bf n} = 0$ on
$\partial\Omega$, and $\nabla_h^* \cdot$ is a discrete divergence
operator acting on that space.

The use of discrete divergence in this formulation is compelled by the
well-known problems in the finite element approximation of the Hilbert
space $H(curl,\Omega)\cap H(div,\Omega)$. In particular, in
(\ref{eq1}) we approximate that space conformally with respect to the
curl operator. This yields a linear system in the form of a Hodge
Laplacian discretized using edge (curl-conforming) elements. We
demonstrate that this system can be solved using a multigrid method
derived from the previous work of Bochev, Hu, Siefert and Tuminaro
\cite{AMG:2007}. A complementary least-squares formulation wherein
$H(curl,\Omega)\cap H(div,\Omega)$ is approximated by div-conforming
elements yields a linear system corresponding to a Hodge Laplacian
discretized by face (div-conforming) elements. To solve this system we
use an alternative formulation of the multigrid method derived in
\cite{AMG:2008}.

\begin{thebibliography}{99}
\bibitem{AMG:2007} P.~Bochev, J.~Hu, C.~Siefert, and R.~Tuminaro.
\newblock An algebraic multigrid approach based on a compatible gauge
reformulation of {M}axwell's equations.
\newblock {\em SIAM J. Sci. Comp.} Volume 31, Issue 1, pp. 557-0583, 2008.

\bibitem{AMG:2008} P.~Bochev, C.~Siefert, R.~Tuminaro, J.~Xu, and Y.~Zhu,
Compatible Gauge Approaches for $H(div,\Omega)$ Equations. CSRI Summer
proceedings, Sandia National Laboratories Report SAND-7977, pp. 3-14,
2007.
\end{thebibliography}

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