Convergence and optimality of adaptive edge finite element methods for time-harmonic Maxwell equations

We consider a standard Adaptive Edge Finite Element Method (AEFEM) based on arbitrary order Nedelec edge elements, for three-dimensional indefinite time-harmonic Maxwell equations. We prove that the AEFEM gives a contraction for the sum of the energy error and the scaled error estimator, between two consecutive adaptive loops provided the initial mesh is fine enough. Using the geometric decay, we show that the AEFEM yields the best-possible decay rate of the error plus oscillation in terms of the number of degrees of freedom. The main technical contribution of the paper is in the establishment of a quasi-orthogonality and a localized a posteriori error estimator.

[1]  Serge Nicaise,et al.  Singularities of eddy current problems , 2003 .

[2]  William F. Mitchell,et al.  A comparison of adaptive refinement techniques for elliptic problems , 1989, TOMS.

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

[4]  Jinchao Xu,et al.  Nodal Auxiliary Space Preconditioning in H(curl) and H(div) Spaces , 2007, SIAM J. Numer. Anal..

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

[6]  D. Arnold,et al.  Finite element exterior calculus, homological techniques, and applications , 2006, Acta Numerica.

[7]  J. Schöberl.  CONVERGENCE OF ADAPTIVE EDGE ELEMENT METHODS FOR THE 3 D EDDY CURRENTS EQUATIONS , 2007 .

[8]  Ricardo H. Nochetto,et al.  Optimal multilevel methods for H(grad), H(curl), and H(div) systems on graded and unstructured grids , 2009 .

[9]  R. Hiptmair.  Multigrid Method for Maxwell's Equations , 1998 .

[10]  J. Nédélec.  A new family of mixed finite elements in ℝ3 , 1986 .

[11]  R. Hiptmair,et al.  Local Multigrid in H(curl) , 2009, 0901.0764.

[12]  Carsten Carstensen,et al.  Convergence analysis of an adaptive edge finite element method for the 2D eddy current equations , 2005, J. Num. Math..

[13]  Rob P. Stevenson,et al.  Sparse Tensor Product Wavelet Approximation of Singular Functions , 2010, SIAM J. Math. Anal..

[14]  Alberto Valli,et al.  An optimal domain decomposition preconditioner for low-frequency time-harmonic Maxwell equations , 1999, Math. Comput..

[15]  R. Nochetto,et al.  Theory of adaptive finite element methods: An introduction , 2009 .

[16]  Joseph E. Pasciak,et al.  Analysis of a Multigrid Algorithm for Time Harmonic Maxwell Equations , 2004, SIAM J. Numer. Anal..

[17]  Peter Monk,et al.  A posteriori error indicators for Maxwell's equations , 1998 .

[18]  M. Costabel,et al.  Singularities of Electromagnetic Fields¶in Polyhedral Domains , 2000 .

[19]  Ricardo H. Nochetto,et al.  Data Oscillation and Convergence of Adaptive FEM , 2000, SIAM J. Numer. Anal..

[20]  A. Robinson.  I. Introduction , 1991 .

[21]  Peter Monk,et al.  A finite element method for approximating the time-harmonic Maxwell equations , 1992 .

[22]  Hoppe,et al.  CONVERGENCE OF ADAPTIVE EDGE ELEMENT METHODS FOR THE 3D EDDY CURRENTS EQUATIONS , 2009 .

[23]  Igor Kossaczký.  A recursive approach to local mesh refinement in two and three dimensions , 1994 .

[24]  Ferenc Izsák,et al.  A reliable and efficient implicit a posteriori error estimation technique for the time harmonic Maxwell equations , 2007 .

[25]  P. Deuflhard,et al.  Adaptive Multilevel Methods for Edge Element Discretizations of Maxwell's Equations , 1997 .

[26]  Zhong LiuQiang.  Fast algorithms of edge element discretizations and adaptive finite element methods for two classes of Maxwell equations , 2013 .

[27]  ROB STEVENSON,et al.  The completion of locally refined simplicial partitions created by bisection , 2008, Math. Comput..

[28]  R. Hoppe,et al.  Residual based a posteriori error estimators for eddy current computation , 2000 .

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

[30]  Snorre H. Christiansen,et al.  Smoothed projections in finite element exterior calculus , 2007, Math. Comput..

[31]  J. Nédélec.  Mixed finite elements in ℝ3 , 1980 .

[32]  Wolfgang Dahmen,et al.  Adaptive Finite Element Methods with convergence rates , 2004, Numerische Mathematik.

[33]  Jinchao Xu.  A new class of iterative methods for nonselfadjoint or indefinite problems , 1992 .

[34]  P. Monk.  A Simple Proof of Convergence for an Edge Element Discretization of Maxwell’s Equations , 2003 .

[35]  Jinchao Xu,et al.  Iterative Methods by Space Decomposition and Subspace Correction , 1992, SIAM Rev..

[36]  Weiying Zheng,et al.  An Adaptive Multilevel Method for Time-Harmonic Maxwell Equations with Singularities , 2007, SIAM J. Sci. Comput..

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

[38]  Douglas N. Arnold,et al.  Multigrid in H (div) and H (curl) , 2000, Numerische Mathematik.

[39]  Ricardo H. Nochetto,et al.  Convergence of Adaptive Finite Element Methods for General Second Order Linear Elliptic PDEs , 2005, SIAM J. Numer. Anal..

[40]  R. Hiptmair.  Finite elements in computational electromagnetism , 2002, Acta Numerica.

[41]  L. R. Scott,et al.  Finite element interpolation of nonsmooth functions satisfying boundary conditions , 1990 .

[42]  Joseph E. Pasciak,et al.  Overlapping Schwarz preconditioners for indefinite time harmonic Maxwell equations , 2003, Math. Comput..

[43]  Joachim Schöberl,et al.  A posteriori error estimates for Maxwell equations , 2007, Math. Comput..

[44]  Joachim Sch Oberl.  COMMUTING QUASI INTERPOLATION OPERATORS FOR MIXED FINITE ELEMENTS , 2004 .

[45]  Jinchao Xu,et al.  OPTIMAL ERROR ESTIMATES FOR NEDELEC EDGE ELEMENTS FOR TIME-HARMONIC MAXWELL'S EQUATIONS * , 2009 .

[46]  Rob P. Stevenson,et al.  Optimality of a Standard Adaptive Finite Element Method , 2007, Found. Comput. Math..