Stable skeleton integral equations for general coefficient Helmholtz transmission problems
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arXiv
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| Main Authors: | , , |
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| Format: | Preprint |
| Published: |
2025
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| _version_ | 1866918078325981184 |
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| author | Gräßle, Benedikt Hiptmair, Ralf Sauter, Stefan |
| author_facet | Gräßle, Benedikt Hiptmair, Ralf Sauter, Stefan |
| contents | A novel variational formulation of layer potentials and boundary integral operators generalizes their classical construction by Green's functions, which are not explicitly available for Helmholtz problems with variable coefficients. Wavenumber explicit estimates and properties like jump conditions follow directly from their variational definition and enable a non-local (``integral'') formulation of acoustic transmission problems (TP) with piecewise Lipschitz coefficients. We obtain the well-posedness of the integral equations directly from the stability of the underlying TP. The simultaneous analysis for general dimensions and complex wavenumbers (in this paper) imposes an artificial boundary on the external Helmholtz problem and employs recent insights into the associated Dirichlet-to-Neumann map. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2507_00991 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Stable skeleton integral equations for general coefficient Helmholtz transmission problems Gräßle, Benedikt Hiptmair, Ralf Sauter, Stefan Analysis of PDEs Numerical Analysis 31B10, 35C15, 45A05, 65R20 A novel variational formulation of layer potentials and boundary integral operators generalizes their classical construction by Green's functions, which are not explicitly available for Helmholtz problems with variable coefficients. Wavenumber explicit estimates and properties like jump conditions follow directly from their variational definition and enable a non-local (``integral'') formulation of acoustic transmission problems (TP) with piecewise Lipschitz coefficients. We obtain the well-posedness of the integral equations directly from the stability of the underlying TP. The simultaneous analysis for general dimensions and complex wavenumbers (in this paper) imposes an artificial boundary on the external Helmholtz problem and employs recent insights into the associated Dirichlet-to-Neumann map. |
| title | Stable skeleton integral equations for general coefficient Helmholtz transmission problems |
| topic | Analysis of PDEs Numerical Analysis 31B10, 35C15, 45A05, 65R20 |
| url | https://arxiv.org/abs/2507.00991 |