Superconvergence for Control-Volume Mixed Finite Element Methods on Rectangular Grids

We consider control-volume mixed finite element methods for the approximation of second-order elliptic problems on rectangular grids. These methods associate control volumes (covolumes) with the vector variable as well as the scalar, obtaining local algebraic representation of the vector equation (e.g., Darcy’s law) as well as the scalar equation (e.g., conservation of mass). We establish $O(h^2)$ superconvergence for both the scalar variable in a discrete $L^2$-norm and the vector variable in a discrete $H({\rm div})$-norm. The analysis exploits a relationship between control-volume mixed finite element methods and the lowest order Raviart-Thomas mixed finite element methods.

[1]  Jean E. Roberts,et al.  Mixed and hybrid methods , 1991 .

[2]  Philippe G. Ciarlet,et al.  The finite element method for elliptic problems , 2002, Classics in applied mathematics.

[3]  Fabio Milner,et al.  Interior and superconvergence estimates for mixed methods for second order elliptic problems , 1985 .

[4]  Todd Arbogast,et al.  Mixed Finite Element Methods on Nonmatching Multiblock Grids , 2000, SIAM J. Numer. Anal..

[5]  Do Y. Kwak,et al.  Mixed Covolume Methods on Rectangular Grids For Elliptic Problems , 2000, SIAM J. Numer. Anal..

[6]  Michael G. Edwards,et al.  Finite volume discretization with imposed flux continuity for the general tensor pressure equation , 1998 .

[7]  J. Jr. Douglas,et al.  Superconvergence of mixed finite element methods on rectangular domains , 1989 .

[8]  Ricardo G. Durán,et al.  Superconvergence for rectangular mixed finite elements , 1990 .

[9]  M. Wheeler,et al.  Mixed Finite Elements for Elliptic Problems with Tensor Coefficients as Cell-Centered Finite Differences , 1997 .

[10]  T. F. Russell,et al.  Relationships among some locally conservative discretization methods which handle discontinuous coefficients , 2004 .

[11]  Mary F. Wheeler,et al.  Superconvergence of the Velocity in Mimetic Finite Difference Methods on Quadrilaterals , 2005, SIAM J. Numer. Anal..

[12]  I. Yotov,et al.  Mixed Finite Element Methods on Non-Matching Multiblock Grids , 1996 .

[13]  A. H. Schatz,et al.  Interior estimates for Ritz-Galerkin methods , 1974 .

[14]  Michel Fortin,et al.  Mixed and Hybrid Finite Element Methods , 2011, Springer Series in Computational Mathematics.

[15]  Ivar Aavatsmark,et al.  Discretization on Unstructured Grids for Inhomogeneous, Anisotropic Media. Part I: Derivation of the Methods , 1998, SIAM J. Sci. Comput..

[16]  Jim E. Jones,et al.  Control‐volume mixed finite element methods , 1996 .

[17]  P. Raviart,et al.  A mixed finite element method for 2-nd order elliptic problems , 1977 .

[18]  Shirley Dex,et al.  JR 旅客販売総合システム（マルス）における運用及び管理について , 1991 .

[19]  P. Grisvard.  Elliptic Problems in Nonsmooth Domains , 1985 .

[20]  Mary F. Wheeler,et al.  SOME SUPERCONVERGENCE RESULTS FOR MIXED FINITE ELEMENT METHODS FOR ELLIPTIC PROBLEMS ON RECTANGULAR DOMAINS. , 1985 .

[21]  J. David Moulton,et al.  Convergence of mimetic finite difference discretizations of the diffusion equation , 2001, J. Num. Math..

[22]  M. Shashkov,et al.  Mimetic Finite Difference Methods for Diffusion Equations , 2002 .

[23]  T. F. Russell,et al.  Finite element and finite difference methods for continuous flows in porous media. , 1800 .

[24]  Richard E. Ewing,et al.  Superconvergence of the velocity along the Gauss lines in mixed finite element methods , 1991 .

[25]  A. Weiser,et al.  On convergence of block-centered finite differences for elliptic-problems , 1988 .

[26]  Do Y. Kwak,et al.  A General Framework for Constructing and Analyzing Mixed Finite Volume Methods on Quadrilateral Grids: The Overlapping Covolume Case , 2001, SIAM J. Numer. Anal..

[27]  J. D. Wilson,et al.  Shape Functions for Velocity Interpolation in General Hexahedral Cells , 2002 .

[28]  Panayot S. Vassilevski,et al.  A general mixed covolume framework for constructing conservative schemes for elliptic problems , 1999, Math. Comput..