Quasi-optimal convergence rate of an AFEM for quasi-linear problems

We prove the quasi-optimal convergence of a standard adaptive finite element method (AFEM) for nonlinear elliptic second-order equations of monotone type. The adaptive algorithm is based on residual-type a posteriori error estimators and D\"orfler's strategy is assumed for marking. We first prove a contraction property for a suitable definition of total error, which is equivalent to the total error as defined by Casc\'on et al. (in SIAM J. Numer. Anal. 46 (2008), 2524--2550), and implies linear convergence of the algorithm. Secondly, we use this contraction to derive the optimal cardinality of the AFEM.

[1]  Pedro Morin,et al.  Convergence rates for adaptive finite elements , 2008, 0803.3824.

[2]  Pedro Morin,et al.  Convergence and quasi-optimality of adaptive FEM for Steklov eigenvalue problems , 2011 .

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

[4]  Jinchao Xu,et al.  Numerische Mathematik Convergence and optimal complexity of adaptive finite element eigenvalue computations , 2022 .

[5]  Christian Kreuzer,et al.  Linear Convergence of an Adaptive Finite Element Method for the p-Laplacian Equation , 2008, SIAM J. Numer. Anal..

[6]  Ronald A. DeVore,et al.  Multiscale, Nonlinear and Adaptive Approximation: Dedicated to Wolfgang Dahmen on the Occasion of his 60th Birthday , 2009 .

[7]  Eun-Jae Park,et al.  A mixed finite element method for a strongly nonlinear second-order elliptic problem , 1995 .

[8]  Wolfgang Dahmen,et al.  Approximation Classes for Adaptive Methods , 2002 .

[9]  Claudio Padra,et al.  A Posteriori Error Estimators for Nonconforming Approximation of Some Quasi-Newtonian Flows , 1997 .

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

[11]  E. Zeidler.  Nonlinear functional analysis and its applications , 1988 .

[12]  Kunibert G. Siebert,et al.  A BASIC CONVERGENCE RESULT FOR CONFORMING ADAPTIVE FINITE ELEMENTS , 2008 .

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

[14]  Pedro Morin,et al.  Convergence of an adaptive Kačanov FEM for quasi-linear problems , 2010, 1006.3319.

[15]  Nonlinear Monotone Potential Operators: From Nonlinear ODE and PDE to Computational Material Sciences , 2010 .

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

[17]  Stefano Giani,et al.  A Convergent Adaptive Method for Elliptic Eigenvalue Problems , 2009, SIAM J. Numer. Anal..

[18]  Eduardo M. Garau,et al.  Convergence of adaptive finite element methods for eigenvalue problems , 2008, 0803.0365.

[19]  Pekka Neittaanmäki,et al.  Mathematical and Numerical Modelling in Electrical Engineering Theory and Applications , 1996 .

[20]  Eun-Jae Park,et al.  Mixed finite element methods for nonlinear second-order elliptic problems , 1995 .

[21]  S. Chow.  Finite element error estimates for non-linear elliptic equations of monotone type , 1989 .

[22]  P. Concus.  Numerical Solution of the Nonlinear Magnetostatic-Field Equation in Two Dimensions , 1967 .

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

[24]  A. Schmidt,et al.  Design of Adaptive Finite Element Software , 2005 .