Adaptive finite element methods for elliptic problems: Abstract framework and applications
Université de Valenciennes et du Hainaut
Cambrésis, LAMAV, FR CNRS 2956, Institut des Sciences et
Techniques de Valenciennes, 59313 Valenciennes Cedex 9, France.
We consider a general abstract framework of a continuous elliptic problem set on a Hilbert space V that is approximated by a family of (discrete) problems set on a finite-dimensional space of finite dimension not necessarily included into V. We give a series of realistic conditions on an error estimator that allows to conclude that the marking strategy of bulk type leads to the geometric convergence of the adaptive algorithm. These conditions are then verified for different concrete problems like convection-reaction-diffusion problems approximated by a discontinuous Galerkin method with an estimator of residual type or obtained by equilibrated fluxes. Numerical tests that confirm the geometric convergence are presented.
Mathematics Subject Classification: 65N30 / 65N15 / 65N50
Key words: A posteriori estimator / adaptive FEM / discontinuous Galerkin FEM
© EDP Sciences, SMAI, 2010