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Convergence of Adaptive Finite Element Methods

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SIAM Review, 44 (2002), 631--658.
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Pedro Morin

Department of Mathematics

University of Maryland, College Park

College Park, MD 20742, USA

pmorin@math.umd.edu
Ricardo H. Nochetto

Department of Mathematics

University of Maryland, College Park

College Park, MD 20742, USA

rhn@math.umd.edu
Kunibert G.\ Siebert

Institut f\"ur Angewandte Mathematik

Hermann-Herder-Str.\ 10

79104 Freiburg, Germany

kunibert@mathematik.uni-freiburg.de
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Abstract

For quite a long time, adaptive finite element methods have been
widely used in applications. In practice, they converge starting from
coarse grids, although no mathematical theory has been able to prove this
assertion.
Ensuring an error reduction based on a
posteriori error estimators, together with a reduction rate of data
oscillation, we construct a simple and efficient adaptive FEM for
elliptic partial differential equations (PDEs). 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.