A class of Crouzeix-Raviart type nonconforming finite element methods for parabolic variational inequality problem with moving grid on anisotropic meshes

A class of Crouzeix-Raviart type nonconforming finite element methods are proposed for the parabolic variational inequality problem with moving grid on anisotropic meshes. By using some novel approaches and techniques, the same optimal error estimates are obtained as the traditional ones. It is shown that the classical regularity condition or quasi-uniform assumption on meshes is not necessary for the finite element analysis.

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

[2]  Lie-heng Wang,et al.  On the quadratic finite element approximation to the obstacle problem , 2002, Numerische Mathematik.

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

[4]  Yongmin Zhang,et al.  Convergence of free boundaries in discrete obstacle problems , 2007, Numerische Mathematik.

[5]  Zhiming Chen,et al.  On the augmented Lagrangian approach to Signorini elastic contact problem , 2001, Numerische Mathematik.

[6]  G. Stampacchia,et al.  On some non-linear elliptic differential-functional equations , 1966 .

[7]  Peter K. Jimack,et al.  A moving mesh finite element algorithm for the adaptive solution of time-dependent partial differential equations with moving boundaries , 2005 .

[8]  Lie-hengWang.  ON THE ERROR ESTIMATE OF NONCONFORMING FINITE ELEMENT APPROXIMATION TO THE OBSTACLE PROBLEM , 2003 .

[9]  F. T. Suttmeier,et al.  Adaptive computational methods for variational inequalities based on mixed formulations , 2006 .

[10]  Patrick Hild,et al.  Quadratic finite element methods for unilateral contact problems , 2002 .

[11]  Faker Ben Belgacem,et al.  Hybrid finite element methods for the Signorini problem , 2003, Math. Comput..

[12]  Zhang,et al.  A NONCONFORMING ANISOTROPIC FINITE ELEMENT APPROXIMATION WITH MOVING GRIDS FOR STOKES PROBLEM , 2006 .

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

[14]  Andreas Krebs,et al.  A p-version finite element method for nonlinear elliptic variational inequalities in 2D , 2006, Numerische Mathematik.

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

[16]  Lie-heng Wang.  Error estimates of two nonconforming finite elements for the obstacle problem , 1986 .

[17]  Dongying Hua,et al.  A mixed finite element method for the unilateral contact problem in elasticity , 2006 .

[18]  William W. Hager,et al.  Error estimates for the finite element solution of variational inequalities , 1977 .

[19]  M. Ainsworth,et al.  A comparison of solvers for linear complementarity problems arising from large-scale masonry structures , 2006 .

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

[21]  Dietrich Braess,et al.  A posteriori error estimators for obstacle problems – another look , 2005, Numerische Mathematik.

[22]  Franco Brezzi,et al.  A finite element approximation of a variational inequality related to hydraulics , 1976 .

[23]  Shi Dong-yang.  Anisotropic Nonconforming Finite Element Approximation to Variational Inequality Problems with Displacement Obstacle , 2006 .

[24]  Qun Lin,et al.  An immediate analysis for global superconvergence for integrodifferential equations , 1997 .

[25]  I. Babuska,et al.  ON THE ANGLE CONDITION IN THE FINITE ELEMENT METHOD , 1976 .

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

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

[28]  P. Dechaumphai,et al.  Nodeless variable finite element method for heat transfer analysis by means of flux-based formulation and mesh adaptation , 2006 .