In this paper, we construct some $H(curl^2)$-conforming finite elements on a rectangle (a parallelogram) and a triangle. The proposed elements possess some nice properties which have been proved by a rigorous theoretical analysis. Then we apply our new elements to construct a finite element space for discretizing the quad-curl problem. Convergence orders $O(h^k)$ in the $H$(curl) norm and $O(h^k-1)$ in the $H(curl^2)$ norm are established. Numerical experiments are provided to confirm the theoretical results.
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%0 Journal Article
%1 zhang2019hcurltextdollar2textdollarconforming
%A Zhang, Qian
%A Wang, Lixiu
%A Zhang, Zhimin
%D 2019
%I Society for Industrial & Applied Mathematics (SIAM)
%J SIAM Journal on Scientific Computing
%K 35b45-pdes-a-priori-estimates 35q60-pdes-in-connection-with-optics-and-electromagnetic-theory 65n12-pdes-bvps-stability-and-convergence-of-numerical-methods 65n15-pdes-bvps-error-bounds 65n30-pdes-bvps-finite-elements
%N 3
%P A1527--A1547
%R 10.1137/18m1199988
%T H(curl\textdollar\^2\textdollar)-Conforming Finite Elements in 2 Dimensions and Applications to the Quad-Curl Problem
%U https://epubs.siam.org/doi/10.1137/18M1199988
%V 41
%X In this paper, we construct some $H(curl^2)$-conforming finite elements on a rectangle (a parallelogram) and a triangle. The proposed elements possess some nice properties which have been proved by a rigorous theoretical analysis. Then we apply our new elements to construct a finite element space for discretizing the quad-curl problem. Convergence orders $O(h^k)$ in the $H$(curl) norm and $O(h^k-1)$ in the $H(curl^2)$ norm are established. Numerical experiments are provided to confirm the theoretical results.
@article{zhang2019hcurltextdollar2textdollarconforming,
abstract = {In this paper, we construct some $H(curl^2)$-conforming finite elements on a rectangle (a parallelogram) and a triangle. The proposed elements possess some nice properties which have been proved by a rigorous theoretical analysis. Then we apply our new elements to construct a finite element space for discretizing the quad-curl problem. Convergence orders $O(h^k)$ in the $H$(curl) norm and $O(h^{k-1})$ in the $H(curl^2)$ norm are established. Numerical experiments are provided to confirm the theoretical results.},
added-at = {2022-02-17T06:34:46.000+0100},
author = {Zhang, Qian and Wang, Lixiu and Zhang, Zhimin},
biburl = {https://www.bibsonomy.org/bibtex/2cd2d0627243bee54a58532bc62154419/gdmcbain},
doi = {10.1137/18m1199988},
interhash = {b54c7ab29048e1680ba463c675f2499a},
intrahash = {cd2d0627243bee54a58532bc62154419},
journal = {{SIAM} Journal on Scientific Computing},
keywords = {35b45-pdes-a-priori-estimates 35q60-pdes-in-connection-with-optics-and-electromagnetic-theory 65n12-pdes-bvps-stability-and-convergence-of-numerical-methods 65n15-pdes-bvps-error-bounds 65n30-pdes-bvps-finite-elements},
month = jan,
number = 3,
pages = {A1527--A1547},
publisher = {Society for Industrial {\&} Applied Mathematics ({SIAM})},
timestamp = {2022-02-17T06:34:46.000+0100},
title = {H(curl{\textdollar}{\^{}}2{\textdollar})-Conforming Finite Elements in 2 Dimensions and Applications to the Quad-Curl Problem},
url = {https://epubs.siam.org/doi/10.1137/18M1199988},
volume = 41,
year = 2019
}