A meshless method for the Cauchy problem in linear elastodynamics

Abstract

In this paper, we establish new density result for the Navier equation. Based on the denseness of the elastic single-layer potential functions, the Cauchy problem for the Navier equation is investigated. The ill-posedness of this problem is given via the compactness of the operator defined by the potential function. The method combines the Newton’s method and minimum norm solution with discrepancy principle to solve an inverse problem. Convergence and stability estimates are then given with some examples for numerical verification on the efficiency of the proposed method.

Keywords:

• linear elastodynamics
• minimum norm solution
• Cauchy problem
• discrepancy principle

AMS Subject Classifications:

• 65N12
• 65N20
• 65N21
• 65N38

Acknowledgements

We would like to thank the editors and the referees for their careful reading and valuable comments which lead to the improvement of the quality of the submitted manuscript.

Notes

The research was supported by Scientific Research Foundation of CAUC [grant number 2013QD14X] and the Natural Science Foundation of China [grant number 11371172], [grant number 11201476].