Convergence of Adaptive Finite Element Methods

Adaptive finite element methods (FEMs) have been widely used in applications for over 20 years now. In practice, they converge starting from coarse grids, although no mathematical theory has been able to prove this assertion. Ensuring an error reduction rate based on a posteriori error estimators, together with a reduction rate of data oscillation (information missed by the underlying averaging process), we construct a simple and efficient adaptive FEM for elliptic partial differential equations. We prove that this algorithm converges with linear rate without any preliminary mesh adaptation nor explicit knowledge of constants. Any prescribed error tolerance is thus achieved in a finite number of steps. A number of numerical experiments in two and three dimensions yield quasi-optimal meshes along with a competitive performance. Extensions to higher order elements and applications to saddle point problems are discussed as well.

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

[2]  Wolfgang Dahmen,et al.  Adaptive Wavelet Methods for Saddle Point Problems - Optimal Convergence Rates , 2002, SIAM J. Numer. Anal..

[3]  Ricardo H. Nochetto,et al.  Small data oscillation implies the saturation assumption , 2002, Numerische Mathematik.

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

[5]  Reinhard H Stephan Dahlke.  Adaptive Wavelet Methods for Saddle Point Problems , 1999 .

[6]  Wolfgang Dahmen,et al.  Adaptive wavelet methods for elliptic operator equations: Convergence rates , 2001, Math. Comput..

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

[8]  Ricardo H. Nochetto,et al.  An Adaptive Uzawa FEM for the Stokes Problem: Convergence without the Inf-Sup Condition , 2002, SIAM J. Numer. Anal..

[9]  Michael Vogelius,et al.  Feedback and adaptive finite element solution of one-dimensional boundary value problems , 1984 .

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

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

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

[13]  Kunibert G. Siebert,et al.  ALBERT: An adaptive hierarchical nite element toolbox , 2000 .

[14]  Ricardo H. Nochetto,et al.  Local problems on stars: A posteriori error estimators, convergence, and performance , 2003, Math. Comput..

[15]  Kunibert G. Siebert,et al.  ALBERT---Software for scientific computations and applications. , 2001 .

[16]  Willy Dörfler,et al.  An adaptive finite element method for a linear elliptic equation with variable coefficients , 2000 .

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