A gradient recovery technique is proposed and analyzed for finite element solutions which provides new gradient approximations with high order of accuracy. The recovery technique is based on the method of least-squares surface fitting in a finite-dimensional space corresponding to a coarse mesh. It is proved that the recovered gradient has a high order of superconvergence for appropriately chosen surface fitting spaces. The recovery technique is robust, efficient, and applicable to a wide class of problems such as the Stokes and elasticity equations. Key words. finite element methods, superconvergence, error estimates, adaptive refinement AMS subject classifications. 65N30, 65N15, 65F10 PII. S003614290037410X1. Introduction. It has been known for a long time that finite element solutions of partial differential equations can have superconvergence in some subregions of the domain [26,23,8,1]. Superconvergence is a phenomenon that the numerical solution converges to the exact solution at a rate higher than the optimal order error estimate. To exploit superconvergence in the finite element method, several methods have been proposed in the literature in the last 30 years. The method of local averaging has turned out to be a common and useful technique in the study of superconvergence in most of the existing results; see, for example, [23,8,1,7,35,19,18,20,17,21,25,26,10,13] and the references therein. In theory, all the existing results require the underlying finite element mesh to have some special properties such as uniformity [23,7,21], local point symmetry [25,26], local translation invariance [1,26], or orthogonality (e.g., rectangular partition) [8,10,13,19,18,20,28,34].The Zienkiewicz and Zhu (ZZ) method [32, 33] is a procedure which postprocesses the gradient of the finite element solution by using a discrete least-squares fitting on a local patch with high order polynomials. Due to its high efficiency and robustness, the ZZ postprocessing has been widely used for mesh adaptivity and error control in finite element methods [32,33,5,6]. For appropriately chosen discrete norms, this procedure has been computationally justified to yield some superconvergence for the gradient. If the underlying finite element partition is uniform or rectangular, one can provide a theoretical proof for the ZZ method [31,34,30] by using some existing superconvergent estimates [8,23,17,35,18].Our objective of this paper is twofold. First, we modify the ZZ method by applying a global least-squares fitting to the gradient of the finite element approximation. The surface fitting space consists of continuous or discontinuous piecewise polynomials of high order on a coarse partition. Second, we provide a theoretical analysis for

Summary Multipoint-flux-approximation (MPFA) methods were introduced to solve control-volume formulations on general simulation grids for porous-media flow. While these methods are general in the sense that they may be applied to any matching grid, their convergence properties vary. An important property for multiphase flow is the monotonicity of the numerical elliptic operator. In a recent paper (Nordbotten et al. 2007), conditions for monotonicity on quadrilateral grids have been developed. These conditions indicate that MPFA formulations that lead to smaller flux stencils are desirable for grids with high aspect ratios or severe skewness and for media with strong anisotropy or strong heterogeneity. The ideas were pursued recently in Aavatsmark et al. (2008), where the L-method was introduced for general media in 2D. For homogeneous media and uniform grids, this method has four-point flux stencils and seven-point cell stencils in two dimensions. The reduced stencils appear as a consequence of adapting the method to the closest neighboring cells. Here, we extend the ideas for discretization on 3D grids, and ideas and results are shown for both conforming and nonconforming grids. The ideas are particularly desirable for simulation grids that contain faults and local grid refinement. We present numerical results herein that include convergence results for single-phase flow on challenging grids in 2D and 3D and for some simple two-phase results. Also, we compare the L-method with the O-method.

fax 01-972-952-9435. AbstractMPFA methods were introduced to solve control-volume formulations on general simulation grids for porous media flow. While these methods are general in the sense that they may be applied to any grid, their convergence properties vary.An important property for multiphase flow is the monotonicity of the numerical elliptic operator. In a recent paper [1], conditions for monotonicity on quadrilateral grids have been developed. These conditions indicate that MPFA formulations which lead to smaller flux stencils are desirable for grids with high aspect ratio or severe skewness and for media with strong anisotropy or strong heterogeneity. The ideas were recently pursued in [2] where the L-method was introduced for general media in 2D. For homogeneous media and uniform grid, this method has four-point flux stencils and seven-point cell stencils in two dimensions. The reduced stencils appear as a consequence of adapting the method to the closest neighboring cellsHere, we extend the ideas for discretization on 3D grids, and ideas and results are shown for both conforming and nonconforming grids. The ideas are particularly desirable for simulation grids which contain faults and local grid refinement.We present numerical results herein which include convergence results for single phase flow on challenging grids in 2D and 3D, and some simple two-phase results. Also, we compare the L-method with the O-method.

This paper presents a meshing algorithm for domains with internal boundaries. It is an extension of the gridding algorithm presented by Persson and Strang. The resulting triangulation matches all boundaries, and the triangles are all nearly equilateral. Equilateral triangles are beneficial for a finite volume discretization, as fluid flow between elements of very different size is only possible at small timesteps. The mesh generator is compared with the well regarded Triangle programme, where both element quality and simulation performance are checked. It is shown that our mesh generator consistently delivers better meshes.

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