The whys and hows of conditioning of DG plane wave Trefftz methods: a single element
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866915500674514944 |
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| author | Coyle, Joseph Nigam, Nilima |
| author_facet | Coyle, Joseph Nigam, Nilima |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2509_14500 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The whys and hows of conditioning of DG plane wave Trefftz methods: a single element Coyle, Joseph Nigam, Nilima Numerical Analysis 65F08, 65N30, 65N22 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. |
| title | The whys and hows of conditioning of DG plane wave Trefftz methods: a single element |
| topic | Numerical Analysis 65F08, 65N30, 65N22 |
| url | https://arxiv.org/abs/2509.14500 |