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CNA Seminar/Colloquium/Joint Pitt-CNA Colloquium

Jeff Ovall
University of Kentucky
Title: A Framework for Constructing and Analyzing Hierarchical-Type Error Estimators

Abstract: Hierarchical bases have a relatively long and productive history within finite element methods, both for linear solvers and a posteriori error estimators. After solving a variational problem in the chosen approximation space $V$, hierarchical error estimators are constructed by (approximately) solving the residual equation in an auxiliary space $W$. For such estimators, efficiency is trivial to prove, but reliability takes more effort. The traditional reliability analysis is restricted to the energy inner-product setting, makes use of a Strong Cauchy Inequality between $V$ and $W$, and assumes that the exact PDE solution is better approximated in $Voplus W$ than it was in $V$ alone---this Saturation Assumption has its critics, but it tends to hold for large body of problems encountered in practice. Because of the approach of the traditional analysis, $W$ tends to be chosen so that $Voplus W$ is one of the standard approximation spaces. We suggest an analysis based on properties of certain quasi-interpolants, which not only avoids the saturation assumption and restrictions to the energy-inner product setting, but also gives a clearer picture of what properties $W$ should have in order for errors to be well-approximated. A practical realization of such an estimato in $RR^3$ will be presented, together with numerical experiments for a variety of problems which illustrate its robustness. Applications to (2D) eigenvalue problems and functional error estimation will also be discussed, if time permits.

Pdf File: OvallCarnegieMellon.pdf
Date: Tuesday, March 29, 2011
Time: 1:20 pm
Location: Wean Hall 7218
Submitted by:  Walkington