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A posteriori error estimator competition for 2nd-order partial differential equations<sup>*</sup>

Friday, December 3, 2010 - 11:00am - 11:45am
Keller 3-180
Carsten Carstensen (Yonsei University)
Five classes of up to 13 a posteriori error estimators compete in three second-order model
cases, namely the conforming and non-conforming first-order approximation of the Poisson-Problem
plus some conforming obstacle problem.
Since it is the natural first step, the error is estimated in the energy norm exclusively
— hence the competition has limited relevance. The competition allows merely guaranteed
error control and excludes the question of the best error guess.
Even nonsmooth problems can be included. For a variational inequality,
Braess considers Lagrange multipliers and some resulting auxiliary equation to view
the a posteriori 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 efficiency for the affine 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 finite element methods.
Our numerical results provide sufficient evidence that guaranteed error control in the
energy norm is indeed possible with efficiency indices between one and two. Furthermore,
accurate error control is slightly more expensive but pays off in all applications under
consideration while adaptive mesh-refinement is sufficiently 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.

References:


  1. S. Bartels, C. Carstensen, R. Klose, An
    experimental survey of a posteriori Courant finite element
    error control for the Poisson equation
    , Adv. Comput. Math., 15
    (2001), pp. 79-106.


  2. C. Carstensen, C. Merdon, Estimator
    competition for Poisson problems
    , J. Comp. Math., 28 (2010),
    pp. 309-330.

  3. C. Carstensen, C. Merdon, Computational
    survey on a posteriori error estimators for nonconforming
    finite element methods, Part I: Poisson problems (in
    preparation)


  4. C. Carstensen, C. Merdon, A posteriori error
    estimator competition for conforming obstacle problems, Part I:
    Theoretical findings (in preparation),
    Part II: Numerical results (in preparation)





* This work was supported by DFG Research Center MATHEON and by the World Class University (WCU) program through the National Research Foundation of Korea (NRF) funded by the
Ministry of Education, Science and Technology R31-2008-000-10049-0.