A Posteriori Error Estimator Competition for Elliptic Pdesin Energy Norms
主 题: A Posteriori Error Estimator Competition for Elliptic Pdesin Energy Norms
报告人: Prof. Carsten Carstensen (Department of Mathematics, Humboldt University of Berlin)
时 间: 2011-10-09 16:00---17:00
地 点: 理科一号楼 1303
Five classes of up to 13 a posteriori error estimators compete in three
second-order model cases, namely the conforming and non-conforming rst-order
approximation of the Poisson-Problem plus some conforming obstacle problem
and non-conforming Crozeix-Raviart FEM for the Stokes problem. Since it is the
natural rst step, the error is estimated in the energy norm exclusively { hence
the competition has limited relevance. The competition allows merely guaran-
teed error control and excludes the question of the best error guess. Even non-
smooth problems can be included. For a variational inequality, Braess considers
Lagrange multipliers and some resulting auxiliary equation to view the a poste-
riori error control of the error in the obstacle problem as computable terms plus
errors and residuals in the auxiliary equation. Hence all the former a posteriori
error estimators apply to this nonlinear benchmark example as well and lead to
surprisingly accurate guaranteed upper error bounds. This approach allows an
extension to more general boundary conditions and a discussion of eciency for
the ane benchmark examples. The Luce-Wohlmuth and the least-square error
estimators win the competition in several computational benchmark problems.
Novel equilibration of nonconsistency residuals and novel conforming averaging
error estimators win the competition for Crouzeix-Raviart nonconforming nite
element methods. Eventually, a novel postprocessing for the compared equilibra-
tion techniques leads to even higher accuracy at low costs. Our numerical results
provide sucient evidence that guaranteed error control in the energy norm is
indeed possible with eciency indices between one and two. Furthermore, accu-
rate error control is slightly more expensive but pays o in all applications under
consideration while adaptive mesh-re nement is suciently pleasant as accurate
when based on explicit residual-based error estimates. Details of our theoretical
and empirical ongoing investigations will be found in the papers quoted below.
