Convergence and superconvergence analysis of a new quadratic Hermite-type triangular element on anisotropic meshes

A new quadratic Hermite-type triangular finite element is conceived to solve a class of two-dimensional second-order elliptic boundary value problems. Its error estimates on anisotropic meshes are developed. Furthermore, we verify that some conditions set to the meshes contribute to the proof of its superconvergence properties, which can improve the approximation results. Numerical examples are given to confirm our theoretical analysis.

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

[2]  Aihui Zhou,et al.  An Analysis of Some High Accuracy Finite Element Methods for Hyperbolic Problems , 2001, SIAM J. Numer. Anal..

[3]  J. Z. Zhu,et al.  The superconvergent patch recovery and a posteriori error estimates. Part 2: Error estimates and adaptivity , 1992 .

[4]  T. Apel.  Anisotropic Finite Elements: Local Estimates and Applications , 1999 .

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

[6]  Shaochun Chen,et al.  Convergence and superconvergence analysis of finite element methods on graded meshes for singularly and semisingularly perturbed reaction-diffusion problems , 2008 .

[7]  Zhong-Ci Shi,et al.  A new superconvergence property of Wilson nonconforming finite element , 1997 .

[8]  Ningning Yan,et al.  Superconvergent derivative recovery for the intermediate finite element family of the second type , 2001 .

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

[10]  M. Zlámal,et al.  Some superconvergence results in the finite element method , 1977 .

[11]  J. Z. Zhu,et al.  The superconvergent patch recovery and a posteriori error estimates. Part 1: The recovery technique , 1992 .

[12]  John E. Osborn,et al.  Superconvergence in the generalized finite element method , 2007, Numerische Mathematik.

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

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