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| Main Authors: | , , |
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| Format: | Preprint |
| Published: |
2026
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2604.13982 |
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| _version_ | 1866915938837725184 |
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| author | Holmen, Daniel Førland Nordbotten, Jan Martin Vatne, Jon Eivind |
| author_facet | Holmen, Daniel Førland Nordbotten, Jan Martin Vatne, Jon Eivind |
| contents | We consider the simplicial de Rham complex and the Čech-de Rham complex, two bigraded Hilbert complexes whose Hodge-Laplace problems govern spatially coupled problems in mixed dimension and homogeneous dimension, respectively. The former complex can be realized as a subcomplex of the latter. In this paper, we quantify how close these complexes are to each other by constructing bounded cochain complexes between them, and thus we quantify how close a mixed-dimensional formulation of a problem is to an equidimensionally coupled formulation of the same problem. From this construction, we derive a priori- and a posteriori error estimates between the associated Hodge-Laplace problems on the two complexes. These estimates represent the error which is introduced by treating a spatially coupled problem as mixed-dimensional, rather than an equidimensional problem with thin overlaps. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_13982 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Approximation properties of double complexes Holmen, Daniel Førland Nordbotten, Jan Martin Vatne, Jon Eivind Numerical Analysis Analysis of PDEs 46C05, 58A12, 58J10 We consider the simplicial de Rham complex and the Čech-de Rham complex, two bigraded Hilbert complexes whose Hodge-Laplace problems govern spatially coupled problems in mixed dimension and homogeneous dimension, respectively. The former complex can be realized as a subcomplex of the latter. In this paper, we quantify how close these complexes are to each other by constructing bounded cochain complexes between them, and thus we quantify how close a mixed-dimensional formulation of a problem is to an equidimensionally coupled formulation of the same problem. From this construction, we derive a priori- and a posteriori error estimates between the associated Hodge-Laplace problems on the two complexes. These estimates represent the error which is introduced by treating a spatially coupled problem as mixed-dimensional, rather than an equidimensional problem with thin overlaps. |
| title | Approximation properties of double complexes |
| topic | Numerical Analysis Analysis of PDEs 46C05, 58A12, 58J10 |
| url | https://arxiv.org/abs/2604.13982 |