Unique continuation for the Lamé system using stabilized finite element methods

• Published in: GEM international journal on geomathematics Vol. 14; no. 1
• Main Authors: Burman, Erik, Preuss, Janosch
• Format: Journal Article
• Language: English
• Published: Berlin/Heidelberg Springer Berlin Heidelberg 01.12.2023
• Subjects: Applications of Mathematics; Computational Science and Engineering; Earth Sciences; Mathematics; Mathematics and Statistics; Original Paper; Conditional Hölder stability; 65N12; Unique continuation; 86-08; 35J15; Finite element methods; 65N30; Lamé system; 65N20
• ISSN: 1869-2672; 1869-2680; 1869-2680
• DOI: 10.1007/s13137-023-00220-1

We introduce an arbitrary order, stabilized finite element method for solving a unique continuation problem subject to the time-harmonic elastic wave equation with variable coefficients. Based on conditional stability estimates we prove convergence rates for the proposed method which take into account the noise level and the polynomial degree. A series of numerical experiments corroborates our theoretical results and explores additional aspects, e.g. how the quality of the reconstruction depends on the geometry of the involved domains. We find that certain convexity properties are crucial to obtain a good recovery of the wave displacement outside the data domain and that higher polynomial orders can be more efficient but also more sensitive to the ill-conditioned nature of the problem.