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arXiv · 2509.14500

The whys and hows of conditioning of DG plane wave Trefftz methods: a single element

Joseph Coyle·Nilima Nigam
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Abstract

Plane-wave Trefftz methods (PWB) for the Helmholtz equation offer significant advantages over standard discretization approaches whose implementation employs more general polynomial basis functions. A disadvantage of these methods is the poor conditioning of the system matrices. In the present paper, we carefully examine the conditioning of the plane-wave discontinuous Galerkin method with reference to a single element. The properties of the mass and stiffness matrices depend on the size and geometry of the element. We study the mass and system matrices arising from a PWB on a single disk-shaped element. We then examine some preconditioning strategies, and present results showing their behaviour with three different criteria: conditioning, the behaviour of GMRES residuals, and impact on the $L^2$-error.

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Joseph Coyle, Nilima Nigam. 2025-09-18. The whys and hows of conditioning of DG plane wave Trefftz methods: a single element. https://arxiv.org/abs/2509.14500

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