Anisotropic Nonconforming $${ EQ}_1^{rot}$$ Quadrilateral Finite Element Approximation to Second Order Elliptic Problems

The main aim of this paper is to study the nonconforming $$EQ_1^{rot}$$ quadrilateral finite element approximation to second order elliptic problems on anisotropic meshes. The optimal order error estimates in broken energy norm and $$L^2$$-norm are obtained, and three numerical experiments are carried out to confirm the theoretical results.

[1]  Serge Nicaise,et al.  Crouzeix-Raviart type finite elements on anisotropic meshes , 2001, Numerische Mathematik.

[2]  Xuejun Xu.  On the Accuracy of Nonconforming Quadrilateral Q1 Element Approximation for the Navier-Stokes Problem , 2000, SIAM J. Numer. Anal..

[3]  Dongwoo Sheen,et al.  Nonconforming Galerkin methods based on quadrilateral elements for second order elliptic problems , 1999 .

[4]  Yuqiu Long,et al.  Generalized conforming quadrilateral membrane element with vertex rigid rotational freedom , 1994 .

[5]  Dongyang Shi,et al.  ACCURACY ANALYSIS FOR QUASI-WILSON ELEMENT , 2000 .

[6]  Q. Lin,et al.  Superconvergence and extrapolation of non-conforming low order finite elements applied to the Poisson equation , 2005 .

[7]  Dongyang Shi,et al.  LOW ORDER CROUZEIX-RAVIART TYPE NONCONFORMING FINITE ELEMENT METHODS FOR APPROXIMATING MAXWELL'S EQUATIONS , 2008 .

[8]  Wang,et al.  AN ANISOTROPIC NONCONFORMING FINITE ELEMENT METHOD FOR APPROXIMATING A CLASS OF NONLINEAR SOBOLEV EQUATIONS , 2009 .

[9]  Dongyang Shi,et al.  Anisotropic interpolations with application to nonconforming elements , 2004 .

[10]  Zhongci Shi.  A convergence condition for the quadrilateral Wilson element , 1984 .

[11]  Thomas Apel,et al.  Anisotropic interpolation with applications to the finite element method , 1991, Computing.

[12]  Eugene L. Wachspress,et al.  Incompatible quadrilateral basis functions , 1978 .

[13]  Jincheng Ren,et al.  Nonconforming mixed finite element approximation to the stationary Navier–Stokes equations on anisotropic meshes , 2009 .

[14]  Shipeng Mao,et al.  A locking-free anisotropic nonconforming finite element for planar linear elasticity problem , 2007 .

[15]  Dongyang Shi,et al.  A class of Crouzeix-Raviart type nonconforming finite element methods for parabolic variational inequality problem with moving grid on anisotropic meshes , 2007 .

[16]  Shipeng Mao,et al.  CONVERGENCE AND SUPERCONVERGENCE OF A NONCONFORMING FINITE ELEMENT ON ANISOTROPIC MESHES , 2007 .

[17]  Shao-chunChen,et al.  AN ANISOTROPIC NONCONFORMING FINITE ELEMENT WITH SOME SUPERCONVERGENCE RESULTS , 2005 .

[18]  Thomas Apel,et al.  Anisotropic interpolation error estimates for isoparametric quadrilateral finite elements , 1998, Computing.

[19]  Long Yu-qiu,et al.  A generalized conforming isoparametric element , 1988 .

[20]  R. Rannacher,et al.  Simple nonconforming quadrilateral Stokes element , 1992 .

[21]  Uwe Risch,et al.  Superconvergence of a nonconforming low order finite element , 2005 .

[22]  Dongyang Shi,et al.  Anisotropic interpolation and quasi‐Wilson element for narrow quadrilateral meshes , 2004 .

[23]  Dongwoo Sheen,et al.  P1-Nonconforming Quadrilateral Finite Element Methods for Second-Order Elliptic Problems , 2003, SIAM J. Numer. Anal..

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