A locally conservative least-squares method for Darcy flows.

Least-squares finite-element methods for Darcy flow offer several advantages relative to the mixed-Galerkin method: the avoidance of stability conditions between finite-element spaces, the efficiency of solving symmetric and positive definite systems, and the convenience of using standard, continuous nodal elements for all variables. However, conventional C0 implementations conserve mass only approximately and for this reason they have found limited acceptance in applications where locally conservative velocity fields are of primary interest. In this paper, we show that a properly formulated compatible least-squares method offers the same level of local conservation as a mixed method. The price paid for gaining favourable conservation properties is that one has to give up what is arguably the least important advantage attributed to least-squares finite-element methods: one can no longer use continuous nodal elements for all variables. As an added benefit, compatible least-squares methods inherit the best computational properties of both Galerkin and mixed-Galerkin methods and, in some cases, yield identical results, while offering the advantages of not having to deal with stability conditions and yielding positive definite discrete problems. Numerical results that illustrate our findings are provided. Copyright © 2006 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]