Relaxation property for the adaptivity for ill-posed problems

• Published in: Applicable analysis Vol. 93; no. 2; pp. 223 - 253
• Main Authors: Beilina, Larisa, V. Klibanov, Michael
• Format: Journal Article
• Language: English
• Published: Abingdon Taylor & Francis 01.02.2014; Taylor & Francis Ltd
• Subjects: adaptive finite element method; Approximation; coefficient inverse problem; Coefficients; Computer simulation; Finite element analysis; Finite element method; Ill posed problems; Inverse problems; Matematik; Mathematical analysis; Mathematical sciences; Numerical analysis; numerical studies; relaxation property; Simulation
• ISSN: 0003-6811; 1563-504X; 1563-504X
• DOI: 10.1080/00036811.2013.768339

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.