Relaxation property for the adaptivity for ill-posed problems

Abstract

Adaptive finite element method (adaptivity) is known to be an effective numerical tool for some ill-posed problems. The key advantage of the adaptivity is the image improvement with local mesh refinements. A rigorous proof of this property is the central part of this paper. In terms of coefficient inverse problems with single measurement data, the authors consider the adaptivity as the second stage of a two-stage numerical procedure. The first stage delivers a good approximation of the exact coefficient without an advanced knowledge of a small neighborhood of that coefficient. This is a necessary element for the adaptivity to start iterations from. Numerical results for the two-stage procedure are presented for both computationally simulated and experimental data.

Keywords:

• adaptive finite element method
• relaxation property
• ill-posed problems
• coefficient inverse problem
• numerical studies

AMS Subject Classifications:

• 35L10
• 35K10
• 94A40

Acknowledgments

This research was supported by US Army Research Laboratory and US Army Research Office grant W911NF-11-1-0399, the Swedish Research Council (VR), the Swedish Foundation for Strategic Research (SSF) through the Gothenburg Mathematical Modelling Centre (GMMC) and by the Swedish Institute, Visby Program.