Volume 14, Issue 5
Approximation of H(div) with High-Order Optimal Finite Elements for Pyramids, Prisms and Hexahedra

Commun. Comput. Phys., 14 (2013), pp. 1372-1414.

Published online: 2013-11

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• Abstract

Classical facet elements do not provide an optimal rate of convergence of the numerical solution toward the solution of the exact problem in $H(div)$-norm for general unstructured meshes containing hexahedra and prisms. We propose two new families of high-order elements for hexahedra, triangular prisms and pyramids that recover the optimal convergence. These elements have compatible restrictions with each other, such that they can be used directly on general hybrid meshes. Moreover, the $H(div)$ proposed spaces are completing the De Rham diagram with optimal elements previously constructed for $H^1$ and $H(curl)$ approximation. The obtained pyramidal elements are compared theoretically and numerically with other elements of the literature. Eventually, numerical results demonstrate the efficiency of the finite elements constructed.

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@Article{CiCP-14-1372, author = {}, title = {Approximation of H(div) with High-Order Optimal Finite Elements for Pyramids, Prisms and Hexahedra}, journal = {Communications in Computational Physics}, year = {2013}, volume = {14}, number = {5}, pages = {1372--1414}, abstract = {

Classical facet elements do not provide an optimal rate of convergence of the numerical solution toward the solution of the exact problem in $H(div)$-norm for general unstructured meshes containing hexahedra and prisms. We propose two new families of high-order elements for hexahedra, triangular prisms and pyramids that recover the optimal convergence. These elements have compatible restrictions with each other, such that they can be used directly on general hybrid meshes. Moreover, the $H(div)$ proposed spaces are completing the De Rham diagram with optimal elements previously constructed for $H^1$ and $H(curl)$ approximation. The obtained pyramidal elements are compared theoretically and numerically with other elements of the literature. Eventually, numerical results demonstrate the efficiency of the finite elements constructed.

}, issn = {1991-7120}, doi = {https://doi.org/10.4208/cicp.120712.080313a}, url = {http://global-sci.org/intro/article_detail/cicp/7206.html} }
TY - JOUR T1 - Approximation of H(div) with High-Order Optimal Finite Elements for Pyramids, Prisms and Hexahedra JO - Communications in Computational Physics VL - 5 SP - 1372 EP - 1414 PY - 2013 DA - 2013/11 SN - 14 DO - http://doi.org/10.4208/cicp.120712.080313a UR - https://global-sci.org/intro/article_detail/cicp/7206.html KW - AB -

Classical facet elements do not provide an optimal rate of convergence of the numerical solution toward the solution of the exact problem in $H(div)$-norm for general unstructured meshes containing hexahedra and prisms. We propose two new families of high-order elements for hexahedra, triangular prisms and pyramids that recover the optimal convergence. These elements have compatible restrictions with each other, such that they can be used directly on general hybrid meshes. Moreover, the $H(div)$ proposed spaces are completing the De Rham diagram with optimal elements previously constructed for $H^1$ and $H(curl)$ approximation. The obtained pyramidal elements are compared theoretically and numerically with other elements of the literature. Eventually, numerical results demonstrate the efficiency of the finite elements constructed.

Morgane Bergot & Marc Duruflé. (2020). Approximation of H(div) with High-Order Optimal Finite Elements for Pyramids, Prisms and Hexahedra. Communications in Computational Physics. 14 (5). 1372-1414. doi:10.4208/cicp.120712.080313a
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