Estimates of errors generated by uncertain data in a coupled pieso-electric problem

S. Repin ; T. Samrowski

Zapiski Nauchnykh Seminarov POMI, Investigations on applied mathematics and informatics. Part III, Tome 539 (2024), pp. 157-179

Résumé

The paper is concerned with a coupled piezo-electric problem with incompletely known coefficients of the elasticity tensor and two other tensors that define electric properties of the media. Due to this uncertainty, the problem possesses a set (cloud) of equally probable solutions instead of the unique solution. Quantitative characteristics of this set are derived by a posteriori estimates of the functional type. They give an upper bound of the cloud diameter and lower bound of maximal diameter of the ball inscribed. The estimates are fully computable. They are based on solving algebraic optimisation problems of low dimensionality related to the sets containing possible coefficients. In the case of isotropic elasticity with the Poisson's ratio close to 0.5, it is shown that even tiny values of uncertainty in the coefficient may generate very large errors in the solution.