Nonstandard Finite Element Methods

Abstract

The workshop \emph{Nonstandard Finite Element Methods}, organized by Susanne C. Brenner (Baton Rouge), Carsten Carstensen (Berlin) and Peter Monk (Newark) was held August 10 -- 16, 2008. This meeting was well attended with over 40 participants with broad geographic representation. Although Courant is often credited with the discovery of finite elements \cite{cia78,courant43}, the first practical use of such methods in engineering seems to date back to the work of Argyris and Clough et. al. (apparently Clough coined the name \`\`finite elements'') in the 1950s \cite{arg54,tur56}. By the 1970s the finite element method was firmly established in engineering practice and the basic theory of conforming elements for elliptic problems was well understood \cite{cia78}. Engineers and mathematicians have since expanded the use of finite elements to a wide variety of new applications and have improved the theoretical underpinnings of the method. In addition, efficient computational algorithms have been developed, such as the multigrid method, that allow for the application of finite elements to sophisticated three dimensional problems. Within current research, we can distinguish classical or standard finite element methods, and newer nonstandard finite elements The former include conforming elements applied to\`\`standard" Galerkin discretizations of problems. Usually accuracy is obtained by mesh refinement. As the range of applications has increased, and to alleviate perceived shortcomings in standard conforming elements, numerical analysts and engineers have sought more general finite element methods based, for example, on mixed formulations, and non-conforming or even discontinuous Galerkin schemes. These methods may be superior to standard finite elements in particular applications because of enhancements to stability, robustness or conservation properties. We refer to them as non-standard finite element methods. This Oberwolfach workshop was devoted to non-standard finite element methods and their analyses. Inevitability such a snapshot of the subject is biased, but we have tried to include non-standard methods in the broadest sense including mixed, discontinuous Galerkin, generalized, partition of unity and mortar FEM methods. All these schemes have in common that stability and convergence is not obvious and requires mathematical analysis. This is even more true for developing fast solvers, a posteriori error estimation and adaptive mesh design. Through the week there were 29 presentations on various non-standard topics. These provided an overview of current research directions, new developments and open problems. \begin{thebibliography}{1} \bibitem{arg54} {\sc J.\~Argyris}, {\em Energy theorems and structural analysis, part {I}: {G}eneral theorey}, Aircraft Engineering, 26 (1954), pp.\~347--56, 383--87, 394. \bibitem{cia78} {\sc P.\~Ciarlet}, {\em The Finite Element Method for Elliptic Problems}, vol.\~4 of Studies In Mathematics and Its Applications, North-Holland, New York, 1978. \bibitem{courant43} {\sc R.\~Courant}, {\em Variational methods for the solution of problems of equilibrium and vibrations}, Bull. Amer. Math. Soc., 49 (1943), pp.\~1--23. \bibitem{tur56} {\sc M.\~Turner, R.\~Clough, H.\~Martin, and L.\~Topp}, {\em Stiffness and deflection analysis of complex structures}, J. Aero. Sci., 23 (1956), pp.\~805--23. \end{thebibliography}