===affil2:
Sandia National Laboratories
===firstname:
Christopher
===firstname4:

===firstname3:
Oksana
===lastname2:
Bochev
===lastname:
Siefert
===firstname5:

===affil6:

===lastname3:
Guba
===email:
csiefer@sandia.gov
===lastname6:

===affil5:

===otherauths:

===lastname4:

===affil4:

===lastname7:

===affil7:

===firstname7:

===postal:
Sandia National Laboratories, P.O. Box 5800, MS 1323, Albuquerque, NM 87185-1323
===firstname6:

===ABSTRACT:
We present a mimetic least squares method for Darcy flow,
using nodal elements for pressure and lowest order face elements for the fluxes.  Because the pressure and flux are not subject to a joint inf-sup condition, we can choose the discrete spaces independently.  The choice of a face element discretization for the fluxes gives us a method with improved stability, accuracy and conservation properties.

The mimetic least-squares functional is norm-equivalent to a norm on $H^1(\Omega)\times H(div)$. As a result, the associated linear system, though positive definite, has a 2x2 block structure, where one diagonal element is a nodal Laplacian, and the other is a grad-div problem.  Owing to the norm-equivalence of the least-squares functional, the system can be effectively preconditioned by its diagonal.

We treat the least-squares algebraic equations with smoothed
aggregation algebraic multigrid and the latter with the compatible
gauge approach of Bochev et al. [1].  The structure of the equations allows us to combine these into a simple block preconditioner.  We demonstrate good algorithmic and parallel scalability of our problem on up to 17,000 cores on NERSC's hopper machine.

[1] P. Bochev, C. Siefert, R. Tuminaro, J. Xu and Y. Zhu. Compatible Gauge Approaches for H(div) Equations. Technical Report, SAND 2007-5384P, Sandia National Laboratories, August 2007.
===affil3:
Sandia National Laboratories
===lastname5:

===affilother:

===title:
Mimetic Least Squares Methods with Preconditioners for Darcy Flow
===firstname2:
Pavel