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Bogner–Fox–Schmitt
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| Degrees | \(k=3\) where \(k\) is the polynomial subdegree |
| Polynomial subdegree | \(k\) |
| Polynomial superdegree | \(2k\) |
| Lagrange subdegree | \(k\) |
| Lagrange superdegree | \(k\) |
| Reference cells | quadrilateral |
| Finite dimensional space | \(\mathcal{Q}_{k}\) ↓ Show set definitions ↓ |
| DOFs | On each vertex: point evaluations, point evaluations of derivatives in coordinate directions, and point evaluation of mixed second derivative |
| Number of DOFs | quadrilateral: \(16\) |
| Mapping | see [2] |
| continuity | Function values and derivatives are continuous. |
| Categories | Scalar-valued elements |
Implementations
This element is implemented in Symfem .↓ Show implementation detail ↓Examples
| quadrilateral degree 3 |  (click to view basis functions) |
References
- [1] Bogner, F. K., Fox, R. L., and Schmit, L. A. The generation of interelement compatible stiffness and mass matrices by the use of interpolation formulae, Proceedings of the Conference on Matrix Methods in Structural Mechanics, 397–444, 1965. [BibTeX]
- [2] Kirby, Robert C. A general approach to transforming finite elements, The SMAI journal of computational mathematics 4, 197–224, 2018. [DOI: 10.5802/smai-jcm.33] [BibTeX]
DefElement stats
| Element added | 06 March 2021 |
| Element last updated | 20 November 2025 |