GMRES: a generalized minimal residual algorithm for solving nonsymmetric linear systems
SIAM Journal on Scientific and Statistical Computing
CGS, a fast Lanczos-type solver for nonsymmetric linear systems
SIAM Journal on Scientific and Statistical Computing
A Theory for Multiresolution Signal Decomposition: The Wavelet Representation
IEEE Transactions on Pattern Analysis and Machine Intelligence
Ten lectures on wavelets
SIAM Journal on Scientific and Statistical Computing
Parallel Preconditioning with Sparse Approximate Inverses
SIAM Journal on Scientific Computing
SIAM Journal on Numerical Analysis
Accuracy and Stability of Numerical Algorithms
Accuracy and Stability of Numerical Algorithms
Preconditioning techniques for large linear systems: a survey
Journal of Computational Physics
Iterative Methods for Sparse Linear Systems
Iterative Methods for Sparse Linear Systems
The Ultra-Weak Variational Formulation for Elastic Wave Problems
SIAM Journal on Scientific Computing
Preconditioning techniques for the solution of the Helmholtz equation by the finite element method
Mathematics and Computers in Simulation - Special issue: Wave phenomena in physics and engineering: New models, algorithms, and appications
Plane wave decomposition in the unit disc: convergence estimates and computational aspects
Journal of Computational and Applied Mathematics
An ultra-weak method for acoustic fluid-solid interaction
Journal of Computational and Applied Mathematics
Hi-index | 31.45 |
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.