INTERNATIONAL JOURNAL OF c  2010 Institute for Scientiﬁc NUMERICAL ANALYSIS AND MODELING Computing and Information Volume 7, Number 4, Pages 656–666 LOCAL PROJECTION FINITE ELEMENT STABILIZATION FOR DARCY FLOW KAMEL NAFA Abstract. Local projection based stabilized ﬁnite element methods for the solution of Darcy ﬂow oﬀer several advantages as compared to mixed Galerkin methods. In particular, the avoidance of stability conditions between ﬁnite element spaces, the eﬃciency in solving the reduced linear algebraic system, and the convenience of using equal order continuous approximations for all variables. In this paper we analyze the pressure gradient method for Darcy ﬂow and investigate its stability and convergence properties. Key Words. Stabilized ﬁnite elements, Darcy equations, convergence, error estimates. 1. Introduction Numerical methods for Darcy equations are traditionally-based on a primal sin- gle ﬁeld formulation for the pressure or on the mixed two ﬁeld velocity-pressure formulation. It is well known that the choice of the ﬁnite element spaces, for the mixed formulation, is subject to the inf-sup stability condition ([10]). This has lead to the use of classical mixed Raviart-Thomas and Brezzi-Douglas-Marini ﬁnite elements ([10]). This approach though giving good accuracy for both velocity and pressure ([20]) has its draw back complexity. It has been a few years since stabilized ﬁnite element methods have been ex- tended to the Darcy equations (see, [23], [5], [6], and [12]). Despite the fact that such methods are well established for ﬂuid ﬂow problems based on Stokes-like op- erator (see, [19], [17], [32], [7], [3], [16], [21], and [22]). In [23] a term based on the residual of Darcy law is added to the classical Galerkin formulation making the for- mulation stable for all combination of conforming continuous velocity-pressure ap- proximations. Another class of stabilized methods has been derived using Galerkin methods enriched with bubble functions (see, [1] and [2]). Alternative stabilization techniques based on a least squares formulation have been proposed by ([5]), and ([6]). Recently, local projection methods that seem less sensitive to the choice of pa- rameters and have better local conservation properties were proposed for Stokes problem (see, [14], and [4]). The two-level pressure gradient method with a projec- tion onto a discontinuous ﬁnite element space of a lower degree deﬁned on a coarser grid has been analyzed in [4], [8], [25], [26], and [12]. We note that although the two-level pressure gradient stabilization method gives a slightly bigger discretisa- tion stencil, the drawback is not severe because the pressure-gradient unknowns can be eliminated locally. Received by the editors February 9, 2009 and, in revised form, June 10, 2009. 2000 Mathematics Subject Classiﬁcation. 65N12, 65N30, 65N15, 76D07. This research was supported by Sultan Qaboos University, Project IG/SCI/DOMS/09/12. 656