A posteriori error estimates for nonconforming finite element methods for fourth-order problems on rectangles

The a posteriori error analysis of conforming finite element discretisations of the biharmonic problem for plates is well established, but nonconforming discretisations are more easy to implement in practice. The a posteriori error analysis for the Morley plate element appears very particular because two edge contributions from an integration by parts vanish simultaneously. This crucial property is lacking for popular rectangular nonconforming finite element schemes like the nonconforming rectangular Morley finite element, the incomplete biquadratic finite element, and the Adini finite element. This paper introduces a novel methodology and utilises some conforming discrete space on macro elements to prove reliability and efficiency of an explicit residual-based a posteriori error estimator. An application to the Morley triangular finite element shows the surprising result that all averaging techniques yield reliable error bounds. Numerical experiments confirm the reliability and efficiency for the established a posteriori error control on uniform and graded tensor-product meshes.

[1]  Yuan-Ming Wang,et al.  Time-Delayed finite difference reaction-diffusion systems with nonquasimonotone functions , 2006, Numerische Mathematik.

[2]  Jinchao Xu,et al.  SOME n-RECTANGLE NONCONFORMING ELEMENTS FOR FOURTH ORDER ELLIPTIC EQUATIONS , 2007 .

[3]  T. Gudi.  Residual‐based a posteriori error estimator for the mixed finite element approximation of the biharmonic equation , 2011 .

[4]  L. Morley.  The Triangular Equilibrium Element in the Solution of Plate Bending Problems , 1968 .

[5]  Jun Hu,et al.  A new a posteriori error estimate for the Morley element , 2009, Numerische Mathematik.

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

[7]  Shangyou Zhang,et al.  The Lowest Order Differentiable Finite Element on Rectangular Grids , 2011, SIAM J. Numer. Anal..

[8]  P. Lascaux,et al.  Some nonconforming finite elements for the plate bending problem , 1975 .

[9]  Jun Hu,et al.  A unifying theory of a posteriori error control for nonconforming finite element methods , 2007, Numerische Mathematik.

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

[11]  P. Grisvard.  Singularities in Boundary Value Problems , 1992 .

[12]  Rolf Stenberg,et al.  A posteriori error analysis for the Morley plate element with general boundary conditions , 2010 .

[13]  Paul Houston,et al.  An a posteriori error indicator for discontinuous Galerkin approximations of fourth-order elliptic problems , 2011 .

[14]  Wang Ming,et al.  Nonconforming tetrahedral finite elements for fourth order elliptic equations , 2007, Math. Comput..

[15]  Carsten Carstensen,et al.  Explicit Error Estimates for Courant, Crouzeix-Raviart and Raviart-Thomas Finite Element Methods , 2012 .

[16]  Xuehai Huang,et al.  Convergence of an Adaptive Mixed Finite Element Method for Kirchhoff Plate Bending Problems , 2011, SIAM J. Numer. Anal..

[17]  Carsten Carstensen,et al.  A unifying theory of a posteriori finite element error control , 2005, Numerische Mathematik.

[18]  Rolf Stenberg,et al.  A posteriori error estimates for the Morley plate bending element , 2007, Numerische Mathematik.

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

[20]  Susanne C. Brenner,et al.  A two-level additive Schwarz preconditioner for nonconforming plate elements , 1996 .

[21]  Jun Hu,et al.  Convergence and optimality of the adaptive Morley element method , 2012, Numerische Mathematik.

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

[23]  Long Chen.  FINITE ELEMENT METHOD , 2013 .

[24]  Wang Ming,et al.  The Morley element for fourth order elliptic equations in any dimensions , 2006, Numerische Mathematik.

[25]  Kokou B. Dossou,et al.  A residual-based a posteriori error estimator for the Ciarlet-Raviart formulation of the first biharmonic problem , 1997 .

[26]  S. C. Brenner,et al.  An a posteriori error estimator for a quadratic C0-interior penalty method for the biharmonic problem , 2010 .

[27]  Peter Hansbo,et al.  A Posteriori Error Estimates for Continuous/Discontinuous Galerkin Approximations of the Kirchhoff-Love Plate , 2011 .

[28]  Jun Hu,et al.  Framework for the A Posteriori Error Analysis of Nonconforming Finite Elements , 2007, SIAM J. Numer. Anal..

[29]  J. Z. Zhu,et al.  The finite element method , 1977 .

[30]  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..