Framework for the A Posteriori Error Analysis of Nonconforming Finite Elements

This paper establishes a unified framework for the a posteriori error analysis of a large class of nonconforming finite element methods. The theory assures reliability and efficiency of explicit residual error estimates up to data oscillations under the conditions (H1)-(H2) and applies to several nonconforming finite elements: the Crouzeix-Raviart triangle element, the Han parallelogram element, the nonconforming rotated (NR) parallelogram element of Rannacher and Turek, the constrained NR parallelogram element of Hu and Shi, the $P_1$ element on parallelograms due to Park and Sheen, and the DSSY parallelogram element. The theory is extended to include $1$-irregular meshes with at most one hanging node per edge.

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

[2]  I. Babuska,et al.  A feedback element method with a posteriori error estimation: Part I. The finite element method and some basic properties of the a posteriori error estimator , 1987 .

[3]  P. Clément.  Approximation by finite element functions using local regularization , 1975 .

[4]  Edward L. Wilson,et al.  Incompatible Displacement Models , 1973 .

[5]  G. Fix.  Review: Philippe G. Ciarlet, The finite element method for elliptic problems , 1979 .

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

[7]  R. Durán,et al.  Error estimators for nonconforming finite element approximations of the Stokes problem , 1995 .

[8]  Junhu,et al.  NONCONFORMING QUADRILATERAL ROTATED Q1 ELEMENT FOR REISSNER－MINDLIN PLATE , 2003 .

[9]  Carsten Carstensen,et al.  Each averaging technique yields reliable a posteriori error control in FEM on unstructured grids. Part II: Higher order FEM , 2002, Math. Comput..

[10]  Carsten Carstensen,et al.  A posteriori error estimates for nonconforming finite element methods , 2002, Numerische Mathematik.

[11]  Mark Ainsworth,et al.  Robust A Posteriori Error Estimation for Nonconforming Finite Element Approximation , 2004, SIAM J. Numer. Anal..

[12]  C. Carstensen.  QUASI-INTERPOLATION AND A POSTERIORI ERROR ANALYSIS IN FINITE ELEMENT METHODS , 1999 .

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

[14]  F. Thomasset.  Finite element methods for Navier-Stokes equations , 1980 .

[15]  R. Durán,et al.  A posteriori error estimators for nonconforming finite element methods , 1996 .

[16]  P. Raviart,et al.  Conforming and nonconforming finite element methods for solving the stationary Stokes equations I , 1973 .

[17]  Carsten Carstensen,et al.  Each averaging technique yields reliable a posteriori error control in FEM on unstructured grids. Part I: Low order conforming, nonconforming, and mixed FEM , 2002, Math. Comput..

[18]  Guido Kanschat,et al.  A posteriori error estimates¶for nonconforming finite element schemes , 1999 .

[19]  Carsten Carstensen,et al.  Convergence analysis of an adaptive nonconforming finite element method , 2006, Numerische Mathematik.

[20]  Rüdiger Verfürth,et al.  A posteriori error estimation and adaptive mesh-refinement techniques , 1994 .

[21]  Mark Ainsworth.  A posteriori error estimation for non-conforming quadrilateral finite elements , 2005 .

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

[23]  Rüdiger Verführt,et al.  A review of a posteriori error estimation and adaptive mesh-refinement techniques , 1996, Advances in numerical mathematics.

[24]  Shi,et al.  CONSTRAINED QUADRILATERAL NONCONFORMING ROTATED Q1 ELEMENT , 2005 .

[25]  L. R. Scott,et al.  Finite element interpolation of nonsmooth functions satisfying boundary conditions , 1990 .

[26]  Zhongci Shi.  The F-E-M test for convergence of nonconforming finite elements , 1987 .

[27]  V. Girault,et al.  A Local Regularization Operator for Triangular and Quadrilateral Finite Elements , 1998 .

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

[29]  Douglas N. Arnold,et al.  Approximation by quadrilateral finite elements , 2000, Math. Comput..