A Recovery Based Linear Finite Element Method For 1D Bi-Harmonic Problems

We analyze a gradient recovery based linear finite element method to solve bi-harmonic equations and the corresponding eigenvalue problems. Our method uses only $$C^0$$C0 element, which avoids complicated construction of $$C^1$$C1 elements and nonconforming elements. Optimal error bounds under various Sobolev norms are established. Moreover, after a post-processing the recovered gradient is superconvergent to the exact one. Some numerical experiments are presented to validate our theoretical findings. As an application, the new method has been also used to solve 1-D fully nonlinear Monge–Ampère equation numerically.

[1]  Mohamed El-Gamel,et al.  An Efficient Technique for Finding the Eigenvalues of Fourth-Order Sturm-Liouville Problems , 2012 .

[2]  Ahmed Naga,et al.  THE POLYNOMIAL-PRESERVING RECOVERY FOR HIGHER ORDER FINITE ELEMENT METHODS IN 2D AND 3D , 2005 .

[3]  F. Chatelin.  Spectral approximation of linear operators , 2011 .

[4]  Zhimin Zhang,et al.  Function Value Recovery and Its Application in Eigenvalue Problems , 2012, SIAM J. Numer. Anal..

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

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

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

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

[9]  Bishnu P. Lamichhane,et al.  A finite element method for a biharmonic equation based on gradient recovery operators , 2014 .

[10]  J. Z. Zhu,et al.  Superconvergence recovery technique and a posteriori error estimators , 1990 .

[11]  Zhimin Zhang,et al.  Analysis of recovery type a posteriori error estimators for mildly structured grids , 2003, Math. Comput..

[12]  J. Oden,et al.  A Posteriori Error Estimation in Finite Element Analysis , 2000 .

[13]  J. Oden,et al.  A Posteriori Error Estimation in Finite Element Analysis: Oden/A Posteriori , 2000 .

[14]  Zhimin Zhang,et al.  A Posteriori Error Estimates Based on the Polynomial Preserving Recovery , 2004, SIAM J. Numer. Anal..

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

[16]  Do Y. Kwak,et al.  Immersed finite element method for eigenvalue problem , 2014, J. Comput. Appl. Math..

[17]  Zhimin Zhang,et al.  A New Finite Element Gradient Recovery Method: Superconvergence Property , 2005, SIAM J. Sci. Comput..