\(H^1, H(\text{curl})\) and \(H(\text{div})\) conforming elements on polygon-based prisms and cones. (English) Zbl 1446.65163
Summary: The conation and extrusion techniques were proposed by A. Bossavit [Math. Comput. Simul. 80, No. 8, 1567–1577 (2010; Zbl 1196.78024)] for constructing \((m+1)\)-dimensional Whitney forms on prisms/cones from \(m\)-dimensional ones defined on the base shape. We combine the conation and extrusion techniques with the 2D polygonal \(H(\text{div})\) conforming finite element proposed by W. Chen and Y. Wang [Math. Comput. 86, No. 307, 2053–2087 (2017; Zbl 1364.65244)], and construct the lowest-order \(H^1, H(\text{curl})\) and \(H(\text{div})\) conforming elements on polygon-based prisms and cones. The elements have optimal approximation rates. Despite of the relatively sophisticated theoretical analysis, the construction itself is easy to implement. As an example, we provide a 100-line Matlab code for evaluating the shape functions of \(H^1, H(\text{curl})\) and \(H(\text{div})\) conforming elements as well as their exterior derivatives on polygon-based cones. Note that all convex and some non-convex 3D polyhedra can be divided into polygon-based cones by connecting the vertices with a chosen interior point. Thus our construction also provides composite elements for all such polyhedra.
MSC:
| 65N30 | Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs |
| 65N50 | Mesh generation, refinement, and adaptive methods for boundary value problems involving PDEs |
Software:
MatlabReferences:
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