Superconvergent biquadratic finite volume element method for two-dimensional Poisson's equations

In this paper, a kind of biquadratic finite volume element method is presented for two-dimensional Poisson's equations by restricting the optimal stress points of biquadratic interpolation as the vertices of control volumes. The method can be effectively implemented by alternating direction technique. It is proved that the method has optimal energy norm error estimates. The superconvergence of numerical gradients at optimal stress points is discussed and it is proved that the method has also superconvergence displacement at nodal points by a modified dual argument technique. Finally, a numerical example verifies the theoretical results and illustrates the effectiveness of the method.

[1]  Yh,et al.  GENERALIZED DIFFERENCE METHODS ON ARBITRARY QUADRILATERAL NETWORKS , 1999 .

[2]  L. R. Scott,et al.  The Mathematical Theory of Finite Element Methods , 1994 .

[3]  Raytcho D. Lazarov,et al.  Error Estimates for a Finite Volume Element Method for Elliptic PDEs in Nonconvex Polygonal Domains , 2004, SIAM J. Numer. Anal..

[4]  Min Yang.  A second-order finite volume element method on quadrilateral meshes for elliptic equations , 2006 .

[5]  Do Y. Kwak,et al.  Lp error estimates and superconvergence for covolume or finite volume element methods , 2003 .

[6]  Jinchao Xu,et al.  Analysis of linear and quadratic simplicial finite volume methods for elliptic equations , 2009, Numerische Mathematik.

[7]  Jim Douglas,et al.  Development and Analysis of Higher Order Finite Volume Methods over Rectangles for Elliptic Equations , 2003, Adv. Comput. Math..

[8]  D. Rose,et al.  Some errors estimates for the box method , 1987 .

[9]  Tongke Wang,et al.  A mixed finite volume element method based on rectangular mesh for biharmonic equations , 2004 .

[10]  Haijun Wu,et al.  Error estimates for finite volume element methods for general second‐order elliptic problems , 2003 .

[11]  Jim Douglas,et al.  ALTERNATING-DIRECTION GALERKIN METHODS ON RECTANGLES , 1971 .

[12]  W. P. Jones,et al.  Analysis of the cell-centred finite volume method for the diffusion equation , 2000 .

[13]  Jim Douglas,et al.  IMPROVED ACCURACY FOR LOCALLY ONE-DIMENSIONAL METHODS FOR PARABOLIC EQUATIONS , 2001 .

[14]  Endre Süli.  Convergence of finite volume schemes for Poisson's equation on nonuniform meshes , 1991 .

[15]  Michael Plexousakis,et al.  On the Construction and Analysis of High Order Locally Conservative Finite Volume-Type Methods for One-Dimensional Elliptic Problems , 2004, SIAM J. Numer. Anal..

[16]  Zhiqiang Cai,et al.  On the accuracy of the finite volume element method for diffusion equations on composite grids , 1990 .

[17]  Tongke Wang.  Alternating direction finite volume element methods for 2D parabolic partial differential equations , 2008 .

[18]  Yunqing Huang,et al.  A SYMMETRIC FINITE VOLUME ELEMENT SCHEME ON QUADRILATERAL GRIDS AND SUPERCONVERGENCE , 2006 .

[19]  V. Thomée,et al.  Error estimates for a finite volume element method for parabolic equations in convex polygonal domains , 2004 .

[20]  Chuanjun Chen,et al.  Error estimation of a quadratic finite volume method on right quadrangular prism grids , 2009 .