Convergence of natural adaptive least squares finite element methods

The first-order div least squares finite element methods provide inherent a posteriori error estimator by the elementwise evaluation of the functional. In this paper we prove Q-linear convergence of the associated adaptive mesh-refining strategy for a sufficiently fine initial mesh with some sufficiently large bulk parameter for piecewise constant right-hand sides in a Poisson model problem. The proof relies on some modification of known supercloseness results to non-convex polygonal domains plus the flux representation formula. The analysis is carried out for the lowest-order case in two-dimensions for the simplicity of the presentation.

[1]  Lei Tang,et al.  Efficiency Based Adaptive Local Refinement for First-Order System Least-Squares Formulations , 2011, SIAM J. Sci. Comput..

[2]  Carsten Carstensen,et al.  Axioms of adaptivity , 2013, Comput. Math. Appl..

[3]  Carsten Carstensen,et al.  Error reduction and convergence for an adaptive mixed finite element method , 2006, Math. Comput..

[4]  Thomas A. Manteuffel,et al.  LOCAL ERROR ESTIMATES AND ADAPTIVE REFINEMENT FOR FIRST-ORDER SYSTEM LEAST SQUARES (FOSLS) , 1997 .

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

[6]  Pavel B. Bochev,et al.  Mathematical Foundations of Least-Squares Finite Element Methods , 2009 .

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

[8]  Jan Brandts,et al.  A note on least-squares mixed finite elements in relation to standard and mixed finite elements , 2006 .

[9]  R. Hoppe,et al.  A review of unified a posteriori finite element error control , 2012 .

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

[11]  Susanne C. Brenner,et al.  Two-level additive Schwarz preconditioners for nonconforming finite element methods , 1996, Math. Comput..

[12]  Christoph Ortner,et al.  Convergence of simple adaptive Galerkin schemes based on h − h/2 error estimators , 2010, Numerische Mathematik.

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

[14]  Carsten Carstensen,et al.  Convergence and Optimality of Adaptive Least Squares Finite Element Methods , 2015, SIAM J. Numer. Anal..

[15]  Gerhard Starke,et al.  An Adaptive Least-Squares Mixed Finite Element Method for Elasto-Plasticity , 2007, SIAM J. Numer. Anal..

[16]  D. Arnold,et al.  Mixed and nonconforming finite element methods : implementation, postprocessing and error estimates , 1985 .

[17]  Ronald A. DeVore,et al.  Fast computation in adaptive tree approximation , 2004, Numerische Mathematik.

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

[19]  Vivette Girault,et al.  Finite Element Methods for Navier-Stokes Equations - Theory and Algorithms , 1986, Springer Series in Computational Mathematics.

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

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

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

[23]  Carsten Carstensen,et al.  Comparison Results of Finite Element Methods for the Poisson Model Problem , 2012, SIAM J. Numer. Anal..

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

[25]  Carsten Carstensen,et al.  Axioms of adaptivity for separate marking , 2016, 1606.02165.

[26]  Carsten Carstensen,et al.  An optimal adaptive mixed finite element method , 2011, Math. Comput..

[27]  L. D. Marini.  An Inexpensive Method for the Evaluation of the Solution of the Lowest Order Raviart–Thomas Mixed Method , 1985 .

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