Center for                           Nonlinear Analysis CNA Home People Seminars Publications Workshops and Conferences CNA Working Groups CNA Comments Form Summer Schools Summer Undergraduate Institute PIRE Cooperation Graduate Topics Courses SIAM Chapter Seminar Positions Contact 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.Recording: http://vnc.math.cmu.edu/cna/CNA-Oval-Mar-29-2011.aviPdf File: OvallCarnegieMellon.pdfDate: Tuesday, March 29, 2011Time: 1:20 pmLocation: Wean Hall 7218Submitted by:  Walkington