Numerical analysis of two plane wave finite element schemes based on the partition of unity method for elastic wave scattering

This work deals with a numerical model based on the partition of unity finite element method (PUFEM), for the two-dimensional time harmonic elastic wave equations. This approach consists in enriching the polynomial finite element spaces locally by superimposed shear (S) and pressure (P) plane waves. Two plane wave Finite elements schemes are particularly investigated here and compared from the point of view of levels of accuracy and conditioning. The problem considered is a horizontal S plane wave scattered by a rigid circular body, in an infinite elastic medium. A finite square domain in the vicinity of the scatterer is considered with the analytical solution of the problem imposed on its boundary through a Robin type boundary condition. An error analysis with respect to the mesh size and the plane wave enrichment is carried out and some numerical aspects, related to the conditioning and its behaviour as a function of the frequency and the number of approximating plane waves, are outlined. The aim is to gain a better practical understanding of the conditioning behaviour of the PUFEM in order to improve its reliability. The results show, for the same number of degrees of freedom per wavelength, higher order elements lead to better accuracy than low order elements. However, this is achieved in detriment of the conditioning especially when the number of degrees of freedom per wavelength is too large.