Robust residual- and recovery-based a posteriori error estimators for interface problems with flux jumps

For elliptic interface problems with flux jumps, this article studies robust residual- and recovery-based a posteriori error estimators for the conforming finite element approximation. The residual estimator is a natural extension of that developed in [Bernardi and Verfurth, Numer Math 85 (2000), 579–608; Petzoldt, Adv Comp Math 16 (2002), 47–75], and the recovery estimator is a nontrivial extension of our method developed in Cai and Zhang, SIAM J Numer Anal 47 (2009) 2132–2156. It is shown theoretically that reliability and efficiency bounds of these error estimators are independent of the jumps provided that the distribution of the coefficients is locally monotone. © 2010 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 28:476–491, 2012

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

[2]  J. Lions,et al.  Non-homogeneous boundary value problems and applications , 1972 .

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

[4]  Michel Fortin,et al.  Mixed and Hybrid Finite Element Methods , 2011, Springer Series in Computational Mathematics.

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

[6]  Rüdiger Verfürth,et al.  A posteriori error estimation and adaptive mesh-refinement techniques , 1994 .

[7]  Shun Zhang,et al.  Discontinuous Galerkin Finite Element Methods for Interface Problems: A Priori and A Posteriori Error Estimations , 2011, SIAM J. Numer. Anal..

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

[9]  Michael J. Holst,et al.  The Finite Element Approximation of the Nonlinear Poisson-Boltzmann Equation , 2007, SIAM J. Numer. Anal..

[10]  Ronald Fedkiw,et al.  The immersed interface method. Numerical solutions of PDEs involving interfaces and irregular domains , 2007, Math. Comput..

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

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

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

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

[15]  W. Marsden.  I and J , 2012 .

[16]  Shun Zhang,et al.  Recovery-Based Error Estimator for Interface Problems: Conforming Linear Elements , 2009, SIAM J. Numer. Anal..

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

[18]  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..

[19]  Shun Zhang,et al.  Recovery-Based Error Estimators for Interface Problems: Mixed and Nonconforming Finite Elements , 2010, SIAM J. Numer. Anal..