SIAM Review, 44 (2002), 631--658.
Pedro Morin
Department of Mathematics
University of Maryland, College Park
College Park, MD 20742, USA
Ricardo H. Nochetto
Department of Mathematics
University of Maryland, College Park
College Park, MD 20742, USA
Kunibert G.\ Siebert
Institut f\"ur Angewandte Mathematik
Hermann-Herder-Str.\ 10
79104 Freiburg, Germany
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.