Past Event: Oden Institute Seminar

Efficient Solution Methodology Based on a Local Wave Tracking Strategy for High Frequency Helmholtz Problems

Rabia Djellouli, Department of Mathematics, California State University Northridge, CSUN

Abstract

We propose a procedure for selecting the direction of the shape functions to improve the efficiency of the solution methodologies that are based on local plane wave approximations. The proposed approach consists in a local wave tracking strategy. More specifically, for each element of the mesh partition (or more larger subdomains), the basis of plane wave is selected such that one plane wave is oriented in the direction of the propagation of the field inside the considered element. The determination of the direction of the field inside the mesh partition is then formulated as a minimization problem. Furthermore, since the resulting minimization problem is nonlinear, it is proposed to apply Newton method to determine the minimum. The computation of the Jacobians and the Hessians that arise at the Newton iterations is based on an exact characterization of the Frechet derivatives of the field with respect to propagation directions. Such a characterization crucial for the stability, fast convergence, and computational efficiency of the Newton algorithm. To illustrate the salient features and highlight the performance of the proposed approach, we present numerical results obtained when incorporating the proposed local wave tracking procedure to the least squares method developed by Monk et al. BIO-SKETCH Rabia Djellouli is Professor of Mathematics and Director of the Interdisciplinary Research Institute for the Sciences (IRIS) at California State University Northridge. He received his Ph.D. in applied mathematics from University of Paris-XI and Ecole Polytechnique, FRANCE. His current research interest includes the mathematical and numerical analysis of wave propagation phenomena, inverse Problems, and some bio-medical engineering applications.