A quasi‐optimal coarse problem and an augmented Krylov solver for the variational theory of complex rays. (10th March 2016)

• Record Type: Journal Article
• Authors: Kovalevsky, Louis; Gosselet, Pierre
• Is Part Of: International journal for numerical methods in engineering. Volume 107:Number 11(2016)
• Page Start: 903
• Page End: 922
• Publication Date: 2016-03-10
• Subjects: discontinuous Galerkin -- Helmholtz equation -- Trefftz method -- variational theory of complex rays -- augmented Krylov solver; Numerical analysis -- Periodicals; Engineering mathematics -- Periodicals; 620.001518
• DOI: 10.1002/nme.5190
• Languages: English
• ISSNs: 0029-5981

Abstract

Summary: The variational theory of complex rays (VTCR) is an indirect Trefftz method designed to study systems governed by Helmholtz‐like equations. It uses wave functions to represent the solution inside elements, which reduces the dispersion error compared with classical polynomial approaches, but the resulting system is prone to be ill‐conditioned. This paper gives a simple and original presentation of the VTCR using the discontinuous Galerkin framework, and it traces back the ill‐conditioning to the accumulation of eigenvalues near zero for the formulation written in terms of wave amplitude. The core of this paper presents an efficient solving strategy that overcomes this issue. The key element is the construction of a search subspace where the condition number is controlled at the cost of a limited decrease of attainable precision. An augmented LSQR solver is then proposed to solve efficiently and accurately the complete system. The approach is successfully applied to different examples. Copyright © 2015 John Wiley & Sons, Ltd.