Two conjectures on the Stokes complex in three dimensions on Freudenthal meshes
<p>In recent years, a great deal of attention has been paid to discretizations of the incompressible Stokes equations that exactly preserve the incompressibility constraint. These are of substantial interest because these discretizations are pressure-robust; i.e., the error estimates for the v...
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| Format: | Journal article |
| Jezik: | English |
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Society for Industrial and Applied Mathematics
2024
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| _version_ | 1826312539734016000 |
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| author | Farrell, PE Mitchell, L Scott, LR |
| author_facet | Farrell, PE Mitchell, L Scott, LR |
| author_sort | Farrell, PE |
| collection | OXFORD |
| description | <p>In recent years, a great deal of attention has been paid to discretizations of the incompressible Stokes equations that exactly preserve the incompressibility constraint. These are of substantial interest because these discretizations are pressure-robust; i.e., the error estimates for the velocity do not depend on the error in the pressure. Similar considerations arise in nearly incompressible linear elastic solids. Conforming discretizations with this property are now well understood in two dimensions but remain poorly understood in three dimensions. In this work, we state two conjectures on this subject. The first is that the Scott–Vogelius element pair is inf-sup stable on uniform meshes for velocity degree <em>k</em>≥4; the best result available in the literature is for <em>k</em>≥6. The second is that there exists a stable space decomposition of the kernel of the divergence for <em>k</em>≥5. We present numerical evidence supporting our conjectures.</p> |
| first_indexed | 2024-03-07T07:57:18Z |
| format | Journal article |
| id | oxford-uuid:46a44f7c-b869-443c-9e5a-9c457fa4fa97 |
| institution | University of Oxford |
| language | English |
| last_indexed | 2024-04-09T03:55:55Z |
| publishDate | 2024 |
| publisher | Society for Industrial and Applied Mathematics |
| record_format | dspace |
| spelling | oxford-uuid:46a44f7c-b869-443c-9e5a-9c457fa4fa972024-03-15T08:19:42ZTwo conjectures on the Stokes complex in three dimensions on Freudenthal meshesJournal articlehttp://purl.org/coar/resource_type/c_dcae04bcuuid:46a44f7c-b869-443c-9e5a-9c457fa4fa97EnglishSymplectic ElementsSociety for Industrial and Applied Mathematics2024Farrell, PEMitchell, LScott, LR<p>In recent years, a great deal of attention has been paid to discretizations of the incompressible Stokes equations that exactly preserve the incompressibility constraint. These are of substantial interest because these discretizations are pressure-robust; i.e., the error estimates for the velocity do not depend on the error in the pressure. Similar considerations arise in nearly incompressible linear elastic solids. Conforming discretizations with this property are now well understood in two dimensions but remain poorly understood in three dimensions. In this work, we state two conjectures on this subject. The first is that the Scott–Vogelius element pair is inf-sup stable on uniform meshes for velocity degree <em>k</em>≥4; the best result available in the literature is for <em>k</em>≥6. The second is that there exists a stable space decomposition of the kernel of the divergence for <em>k</em>≥5. We present numerical evidence supporting our conjectures.</p> |
| spellingShingle | Farrell, PE Mitchell, L Scott, LR Two conjectures on the Stokes complex in three dimensions on Freudenthal meshes |
| title | Two conjectures on the Stokes complex in three dimensions on Freudenthal meshes |
| title_full | Two conjectures on the Stokes complex in three dimensions on Freudenthal meshes |
| title_fullStr | Two conjectures on the Stokes complex in three dimensions on Freudenthal meshes |
| title_full_unstemmed | Two conjectures on the Stokes complex in three dimensions on Freudenthal meshes |
| title_short | Two conjectures on the Stokes complex in three dimensions on Freudenthal meshes |
| title_sort | two conjectures on the stokes complex in three dimensions on freudenthal meshes |
| work_keys_str_mv | AT farrellpe twoconjecturesonthestokescomplexinthreedimensionsonfreudenthalmeshes AT mitchelll twoconjecturesonthestokescomplexinthreedimensionsonfreudenthalmeshes AT scottlr twoconjecturesonthestokescomplexinthreedimensionsonfreudenthalmeshes |