# hpfem/esco2012-boa

Fetching contributors…
Cannot retrieve contributors at this time
54 lines (40 sloc) 2.09 KB
 \title{Worst Case Error Bounds for the Solution of Uncertain Poissson Equations with Mixed Boundary Conditions} \tocauthor{A. Neumaier} \author{} \institute{} \maketitle \begin{center} {\large \underline{Arnold Neumaier}}\\ Faculty of Mathematics, University of Vienna, Austria\\ {\tt Arnold.Neumaier@univie.ac.at} \\ \vspace{4mm}{\large Tanveer Iqbal}\\ Faculty of Mathematics, University of Vienna, Austria\\ {\tt Tanveer.Iqbal@univie.ac.at} \end{center} \section*{Abstract} We consider the solution of linear elliptic partial differential equations with mixed boundary conditions on 2-dimensional domains with a polygonal boundary, not necessarily convex. The equation may contain uncertain parameters constrained by inequalities. We show how to use finite element approximations to compute worst case a posteriori error bounds for linear response functionals determined by the solution. All discretization errors are taken into account. Our bounds are based on the dual weighted residual (DWR) method of {\sc Becker and Rannacher} \cite{BecR}, and treat the uncertainties with the optimization approach described in {\sc Neumaier} \cite{Neu}. To get the error bounds, we use a first order formulation whose solution with linear finite elements produces compatible piecewise linear approximations of the solution and its gradient. For each iteration of the optimization procedure, we need to solve three related boundary value problems, from which we produce the bounds. No knowledge of domain-dependent apriori constants is necessary. We implemented the method for Poisson-type equations with an uncertain mass distribution and mixed Dirichlet/Neumann boundary conditions. \bibliographystyle{plain} \begin{thebibliography}{10} \bibitem{BecR} {\sc R. Becker and R. Rannacher}. {An optimal control approach to a posteriori error estimation in finite element methods}. pp. 1--102 in: Acta Numerica 2001 (A. Iserles, ed.), Cambridge Univ. Press 2001.. \bibitem{Neu} {\sc A. Neumaier}. {Certified error bounds for uncertain elliptic equations}. J. Comput. Appl. Math. 218 (2008), 125--136. \end{thebibliography}