27 research outputs found
Theory and implementation of -matrix based iterative and direct solvers for Helmholtz and elastodynamic oscillatory kernels
In this work, we study the accuracy and efficiency of hierarchical matrix
(-matrix) based fast methods for solving dense linear systems
arising from the discretization of the 3D elastodynamic Green's tensors. It is
well known in the literature that standard -matrix based methods,
although very efficient tools for asymptotically smooth kernels, are not
optimal for oscillatory kernels. -matrix and directional
approaches have been proposed to overcome this problem. However the
implementation of such methods is much more involved than the standard
-matrix representation. The central questions we address are
twofold. (i) What is the frequency-range in which the -matrix
format is an efficient representation for 3D elastodynamic problems? (ii) What
can be expected of such an approach to model problems in mechanical
engineering? We show that even though the method is not optimal (in the sense
that more involved representations can lead to faster algorithms) an efficient
solver can be easily developed. The capabilities of the method are illustrated
on numerical examples using the Boundary Element Method
Finite elements on degenerate meshes: inverse-type inequalities and applications
In this paper we obtain a range of inverse-type inequalities which are applicable to finite-element functions on general classes of meshes, including degenerate meshes obtained by anisotropic refinement. These are obtained for Sobolev norms of positive, zero and negative order. In contrast to classical inverse estimates, negative powers of the minimum mesh diameter are avoided. We give two applications of these estimates in the context of boundary elements: (i) to the analysis of quadrature error in discrete Galerkin methods and (ii) to the analysis of the panel clustering algorithm. Our results show that degeneracy in the meshes yields no degradation in the approximation properties of these method