JournalsowrVol. 1 / No. 2DOI 10.4171/owr/2004/16

Self-Adaptive Methods for PDE

  • Rolf Rannacher

    Universität Heidelberg, Germany
  • Endre Süli

    Oxford University, UK
  • Rüdiger Verfürth

    Ruhr-Universität Bochum, Germany
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This conference was on theoretical and practical aspects of error control and self-adaptivity in the numerical solution of partial differential equations. The organizers were Endre Süli (Oxford), Rolf Rannacher (Heidelberg) and Rüdiger Verfürth (Bochum), and the 44 participants came from 11 different countries. Many processes in sciences and engineering are formulated in terms of partial differential equations. With the incorporation of more and more complete physics these models become increasingly complex and their accurate numerical simulation requires the use of efficient self-adaptive methods. Such ``self-adptivity'' is usually based on {\it a posteriori} estimates for the discretization and iteration errors in terms of local quantities (residuals) obtained from the computed solution. During the last few years, these approaches have seen a rapid development from simple model situations towards real-life applications. The conference's focus was on basic questions of the rigorous mathematical understanding of these methods and their applications. The frame of the conference was set by six invited one-hour survey lectures on the topics: adaptive hp finite element methods, adaptivity in numerical optimization, model adaptivity, convergence of adaptive finite element methods, multiscale adaptivity and wavelets and space-time adaptivity in nonstationary problems. Additionally 18 shorter talks have been given on more specialized aspects not treated in the survey lectures and on other subjects. Important topics addressed by these talks were ``adaptivity for nonlinear, nondifferentiable PDEs'', ``adaptivity for coupled problems'', ``anisotropic and moving mesh adaptation'', ``local and global recovery techniques'', ``parallelization of adaptive methods'', and ``mesh generation''. The talks showed that much progress has been made in extending the concept of a posteriori error estimation and mesh adaptation from simple elliptic problems to real-life situations including coupled models, multiple scales and nonlinearity. Thanks to the restricted number of talks, there was left plenty of time for discussion which was extensively used by the participants. The discussions concentrated on question related to the convergence and asymptotic complexity of adaptive finite element methods and the various strategies of using ``duality-based'' a posteriori error estimates. The recent extension of these concepts to optimization problems and to model calibration has opened a whole new world of applications and will significantly affect the future development of PDE numerics.