Recovery-Based Error Estimator for Interface Problems: Conforming Linear Elements

This paper studies a new recovery-based a posteriori error estimator for the conforming linear finite element approximation to elliptic interface problems. Instead of recovering the gradient in the continuous finite element space, the flux is recovered through a weighted $L^2$ projection onto $H(\mathrm{div})$ conforming finite element spaces. The resulting error estimator is analyzed by establishing the reliability and efficiency bounds and is supported by numerical results. This paper also proposes an adaptive finite element method based on either the recovery-based estimators or the edge estimator through local mesh refinement and establishes its convergence. In particular, it is shown that the reliability and efficiency constants as well as the convergence rate of the adaptive method are independent of the size of jumps.

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

[2]  Gabriel Wittum,et al.  Asymptotically exact a posteriori estimators for the pointwise gradient error on each element in irregular meshes. Part 1: A smooth problem and globally quasi-uniform meshes , 2001, Math. Comput..

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

[4]  Rüdiger Verfürth,et al.  Adaptive finite element methods for elliptic equations with non-smooth coefficients , 2000, Numerische Mathematik.

[5]  Jeffrey S. Ovall,et al.  Fixing a ''Bug'' in Recovery-Type A Posteriori Error Estimators , 2006 .

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

[7]  W. Gibbs,et al.  Finite element methods , 2017, Graduate Studies in Mathematics.

[8]  Jinchao Xu,et al.  Superconvergent Derivative Recovery for Lagrange Triangular Elements of Degree p on Unstructured Grids , 2007, SIAM J. Numer. Anal..

[9]  Christian Kreuzer,et al.  Quasi-Optimal Convergence Rate for an Adaptive Finite Element Method , 2008, SIAM J. Numer. Anal..

[10]  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..

[11]  Jean E. Roberts,et al.  Mixed and hybrid finite element methods , 1987 .

[12]  O. C. Zienkiewicz,et al.  A simple error estimator and adaptive procedure for practical engineerng analysis , 1987 .

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

[14]  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..

[15]  Ningning Yan,et al.  Gradient recovery type a posteriori error estimates for finite element approximations on irregular meshes , 2001 .

[16]  Barbara I. Wohlmuth,et al.  A Local A Posteriori Error Estimator Based on Equilibrated Fluxes , 2004, SIAM J. Numer. Anal..

[17]  Zhimin Zhang.  A Posteriori Error Estimates on Irregular Grids Based on Gradient Recovery , 2001, Adv. Comput. Math..

[18]  Lars B. Wahlbin,et al.  Asymptotically exact a posteriori estimators for the pointwise gradient error on each element in irregular meshes. Part II: The piecewise linear case , 2004, Math. Comput..

[19]  R. Verfürth,et al.  Edge Residuals Dominate A Posteriori Error Estimates for Low Order Finite Element Methods , 1999 .

[20]  I. Babuska,et al.  The finite element method and its reliability , 2001 .

[21]  Carsten Carstensen,et al.  All first-order averaging techniques for a posteriori finite element error control on unstructured grids are efficient and reliable , 2003, Math. Comput..

[22]  Jinchao Xu,et al.  Asymptotically Exact A Posteriori Error Estimators, Part I: Grids with Superconvergence , 2003, SIAM J. Numer. Anal..

[23]  Martin Petzoldt,et al.  A Posteriori Error Estimators for Elliptic Equations with Discontinuous Coefficients , 2002, Adv. Comput. Math..

[24]  Mark Ainsworth.  A posteriori error estimation in the finite element method , 1989 .

[25]  M. Ainsworth,et al.  A posteriori error estimators in the finite element method , 1991 .

[26]  Zhimin Zhang,et al.  Analysis of the superconvergent patch recovery technique and a posteriori error estimator in the finite element method (II) , 1998 .

[27]  Ricardo H. Nochetto,et al.  Convergence of Adaptive Finite Element Methods , 2002, SIAM Rev..

[28]  Zhiming Chen,et al.  On the Efficiency of Adaptive Finite Element Methods for Elliptic Problems with Discontinuous Coefficients , 2002, SIAM J. Sci. Comput..

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

[30]  Carsten Carstensen,et al.  An experimental survey of a posteriori Courant finite element error control for the Poisson equation , 2001, Adv. Comput. Math..

[31]  R. Bruce Kellogg,et al.  On the poisson equation with intersecting interfaces , 1974 .

[32]  W. Dörfler.  A convergent adaptive algorithm for Poisson's equation , 1996 .

[33]  RICARDO H. NOCHETTO,et al.  ADAPTIVE FINITE ELEMENT METHODS FOR ELLIPTIC PDE , .

[34]  Susanne C. Brenner,et al.  Finite Element Methods , 2000 .