Wavelet based ILU preconditioners for the numerical solution by PUFEM of high frequency elastic wave scattering

Daubechies family of wavelets combined to the Incomplete Lower–Upper (ILU) factorization are considered as preconditioners for a block sparse linear system arising from the approximation of the time harmonic elastic wave equations by the Partition of Unity Finite Element Method (PUFEM). After applying the discrete wavelet transform (DWT) to each dense block in the final matrix and the known right-hand side, due to the local enrichment by pressure (P) and shear (S) plane waves, the resulting linear system is solved by the restarted Generalized Minimum RESidual method (GMRES) with ILU preconditioners allowing fill-in elements in the L and U matrix factors. A reordering algorithm of the vertices based on Gibbs method is also introduced. It leads to a significant reduction of the bandwidth of the wave based Finite Element (FE) matrix and enables the ILU preconditioners to be more effective. To study the performance of the proposed preconditioners, a problem of a horizontal S plane wave scattered by a rigid circular body in an infinite elastic medium is considered. The numerical tests show the good performance of the DWT based ILU preconditioners in improving the rate of convergence of GMRES, for high numbers of approximating P and S plane waves, on coarse mesh grids containing multi-wavelength sized elements. Moreover, the Haar DWT enhances the scalability with respect to the problem size, when the number of the nodal points increases, of the ILU preconditioner which uses the threshold strategy in the control of the fill-in elements. Despite the high level of the conditioning, a good accuracy may be achieved for a discretization level of around 1.9 degrees of freedom per S wavelength, which is far below the rule of thumb of 10 nodal points per wavelength, adopted for the conventional FE.