Towards Robust Adaptive Methods for Singularly Perturbed Problems

• Applicant: Professor Dr. Rüdiger Verfürth
• Subject Area: Mathematics
• Term: from 2014 to 2017
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 260814742

• Final Report Year: 2016

Final Report Abstract

We have obtained a fairly complete picture concerning localization results for several Sobolev norms and reaction-diffusion equations with discontinuous diffusion having either dominant reaction or large jumps of the diffusion. These results allow the efficient and robust (with respect to the critical parameters) use of nonlinear tree approximation. We have not obtained our goals concerning instance or rate-optimality of adaptive finite element methods for these equations. This will be at the focus of our future research. Thus, in summary, we have obtained the goals of our proposal concerning nonlinear approximation by finite element functions but not yet concerning adaptive finite element methods.

Publications

• Robust localization of the best error with finite elements in the reaction-diffusion norm. Constr. Approx. 42.2 (2015), pp. 313–347
  F. Tantardini, A. Veeser, R. Verfürth
  (See online at https://doi.org/10.1007/s00365-015-9291-5)