A quasi-optimal coarse problem and an augmented Krylov solver for the Variational Theory of Complex Rays

TL;DR

This paper introduces a new approach to improve the numerical stability of the Variational Theory of Complex Rays by developing a quasi-optimal coarse problem and an augmented Krylov solver, enhancing solution accuracy and efficiency.

Contribution

It presents a novel formulation of VTCR within the discontinuous Galerkin framework and proposes an augmented Krylov solver to address ill-conditioning issues.

Findings

Reduced system ill-conditioning with the new method

Enhanced solver efficiency and accuracy

Successful application to various examples

Abstract

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 to 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.