On the instabilities of naive FEM discretizations for PDEs with sign-changing coefficients
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866915235928997888 |
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| author | Halla, Martin Oberender, Florian |
| author_facet | Halla, Martin Oberender, Florian |
| contents | We consider a scalar diffusion equation with a sign-changing coefficient in its principle part. The well-posedness of such problems has already been studied extensively provided that the contrast of the coefficient is non-critical. Furthermore, many different approaches have been proposed to construct stable discretizations thereof, because naive finite element discretizations are expected to be non-reliable in general. However, no explicit example proving the actual instability is known and numerical experiments often do not manifest instabilities in a conclusive manner. To this end we construct an explicit example with a broad family of meshes for which we prove that the corresponding naive finite element discretizations are unstable. On the other hand, we also provide a broad family of (non-symmetric) meshes for which we prove that the discretizations are stable. Together, these two findings explain the results observed in numerical experiments. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2504_07712 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On the instabilities of naive FEM discretizations for PDEs with sign-changing coefficients Halla, Martin Oberender, Florian Numerical Analysis 65N12, 65N30, 78M10 We consider a scalar diffusion equation with a sign-changing coefficient in its principle part. The well-posedness of such problems has already been studied extensively provided that the contrast of the coefficient is non-critical. Furthermore, many different approaches have been proposed to construct stable discretizations thereof, because naive finite element discretizations are expected to be non-reliable in general. However, no explicit example proving the actual instability is known and numerical experiments often do not manifest instabilities in a conclusive manner. To this end we construct an explicit example with a broad family of meshes for which we prove that the corresponding naive finite element discretizations are unstable. On the other hand, we also provide a broad family of (non-symmetric) meshes for which we prove that the discretizations are stable. Together, these two findings explain the results observed in numerical experiments. |
| title | On the instabilities of naive FEM discretizations for PDEs with sign-changing coefficients |
| topic | Numerical Analysis 65N12, 65N30, 78M10 |
| url | https://arxiv.org/abs/2504.07712 |