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Simple curl-curl-conforming finite elements in two dimensions [article]

Kaibo Hu and Qian Zhang and Zhimin Zhang
2021 arXiv   pre-print
We construct smooth finite element de Rham complexes in two space dimensions. This leads to three families of curl-curl conforming finite elements, two of which contain two existing families.  ...  The simplest triangular and rectangular finite elements have only 6 and 8 degrees of freedom, respectively.  ...  ., constructing simple curl-curl conforming elements with fewer degrees of freedom, compared to, e.g., those in [22] and [21] .  ... 
arXiv:2004.12507v2 fatcat:5fmwck7krrc3ll3tfkszdff5be

New conforming finite element divdiv complexes in three dimensions [article]

Jun Hu, Yizhou Liang, Rui Ma, Min Zhang
2022 arXiv   pre-print
In this paper, the first family of conforming finite element divdiv complexes on cuboid grids in three dimensions is constructed.  ...  Besides, a new family of conforming finite element divdiv complexes with enhanced smoothness on tetrahedral grids is presented.  ...  H(sym curl)-conforming finite element space.  ... 
arXiv:2204.07895v1 fatcat:kxbzqpf4qzhovm3we4hzvh7ooq

The role of the penalty in the local discontinuous Galerkin method for Maxwell's eigenvalue problem

T. Warburton, Mark Embree
2006 Computer Methods in Applied Mechanics and Engineering  
The first set of eigenvalues approximate the spectrum of the curl-conforming finite element curl-curl operator, while the second set are spurious eigenvalues that can be made arbitrarily large by increasing  ...  We analyze the role of this term in the LDG discretization of the Maxwell curl-curl operator, demonstrating that the stabilization parameter can be used to split the spectrum into two sections.  ...  We thank Paul Ledger (ETHZ) for his valuable advice on the construction of curl-conforming finite elements.  ... 
doi:10.1016/j.cma.2005.06.011 fatcat:vza2fzyx3jd4tbiyizvrt2ei74

Nonstandard finite element de Rham complexes on cubical meshes [article]

Andrew Gillette, Kaibo Hu, Shuo Zhang
2018 arXiv   pre-print
We propose two general operations on finite element differential complexes on cubical meshes that can be used to construct and analyze sequences of "nonstandard" convergent methods.  ...  We show that the resulting elements provide convergent, non-conforming methods for problems requiring stronger regularity and satisfy a discrete Korn inequality.  ...  In 2D, our construction leads to simple elements for the Darcy-Stokes-Brinkman problem while in 3D our construction yields the Adini plate element [1] and an H(curl) element that is conforming for Poisson  ... 
arXiv:1804.04390v1 fatcat:xjpqghlssjbtvp6jbgs5mzqnfe

Discrete and conforming smooth de Rham complexes in three dimensions

Michael Neilan
2015 Mathematics of Computation  
In particular, we develop H 2 , H 1 (curl), H 1 and L 2 conforming finite element spaces and show that an exactness property is satisfied.  ...  Conforming discrete de Rham complexes consisting of finite element spaces with extra smoothness are constructed.  ...  Conclusions In this paper, we have constructed conforming finite element spaces in three dimensions with high regularity.  ... 
doi:10.1090/s0025-5718-2015-02958-5 fatcat:reokeqwpv5b53bkd2bxdk6ml2m

Nonconforming finite elements for the Brinkman and -curlΔcurl problems on cubical meshes [article]

Qian Zhang, Min Zhang, Zhimin Zhang
2022 arXiv   pre-print
We propose two families of nonconforming elements on cubical meshes: one for the -curlΔcurl problem and the other for the Brinkman problem.  ...  The element for the Brinkman problem can yield a uniformly stable finite element method with respect to the parameter ν.  ...  In 3D, two of the authors proposed a tetrahedral grad curl-conforming element in [37] , which has 315 degrees of freedom on each element.  ... 
arXiv:2206.08493v2 fatcat:j5p4uq2djbgmljxamiu77mikk4

AN ANALYSIS OF NEW FINITE ELEMENT SPACES FOR MAXWELL'S EQUATIONS
English

Ji Hyun Kim
2018 Journal of Mathematical Sciences. Advances and Applications  
In this paper, we introduce four different finite element spaces on hexahedron for Maxwell's equations.  ...  We also describe interpolation operators that map from suitable subspaces 1 H H H and the space ( ) Ω 2 L into the appropriate finite element spaces.  ...  And the finite element space V ( ) K is conforming in ( curl Ω , H ). Proof.  ... 
doi:10.18642/jmsaa_7100121975 fatcat:hegchthsvjbghnf7day5vcyqhi

Efficient Assembly of $H(\mathrm{div})$ and $H(\mathrm{curl})$ Conforming Finite Elements

Marie E. Rognes, Robert C. Kirby, Anders Logg
2010 SIAM Journal on Scientific Computing  
In this paper, we discuss how to efficiently evaluate and assemble general finite element variational forms on H(div) and H(curl).  ...  The result is an automated, efficient, and easy-to-use implementation that allows a user to specify finite element variational forms on H(div) and H(curl) in close to mathematical notation.  ...  In R 3 , J must also be orthogonal for this to occur. H(div) and H(curl) conforming finite elements.  ... 
doi:10.1137/08073901x fatcat:u4rkyaimsngh7fd5jewc4g54fa

Finite Element Complexes in Two Dimensions [article]

Long Chen, Xuehai Huang
2022 arXiv   pre-print
First smooth scalar finite elements in two dimensions are developed based on a non-overlapping decomposition of the simplicial lattice and the Bernstein basis of the polynomial space.  ...  Dimension count plays an important role for verifying the exactness of two-dimensional finite element complexes.  ...  The de Rham complex and BGG framework are reviewed in Section 2. In Section 3 the geometric decomposition of C m -conforming finite elements in two dimensions is studied.  ... 
arXiv:2206.00851v2 fatcat:wgzldmtmsbdunapr27p37qh7gq

A Family of curl-curl Conforming Finite Elements on Tetrahedral Meshes

Qian Zhang & Zhimin Zhang
2020 CSIAM Transactions on Applied Mathematics  
In this paper, we turn our focus to 3 dimensions and construct H(curl 2 )-conforming finite elements on tetrahedral meshes.  ...  Having the tetrahedral elements, we can solve the quad-curl problem in any Lipschitz domain by the conforming finite element method.  ...  Acknowledgments The research of this work is supported in part by the National Natural Science Foundation of China grants NSFC 11871092 and NASF U1930402.  ... 
doi:10.4208/csiam-am.2020-0023 fatcat:7hr3mevu65f4feb7bwxgkp7s54

A family of finite element Stokes complexes in three dimensions [article]

Kaibo Hu and Qian Zhang and Zhimin Zhang
2021 arXiv   pre-print
In the lowest order case, the finite elements in the complex have 4, 18, 16, and 1 degrees of freedom, respectively.  ...  We construct finite element Stokes complexes on tetrahedral meshes in three-dimensional space.  ...  In particular, any H(curl 2 ; Ω)conforming finite element is automatically H(grad curl; Ω)-conforming.  ... 
arXiv:2008.03793v2 fatcat:z4lv6xlc5zbltdqymtoitt43ei

Curl-curl conforming elements on tetrahedra [article]

Qian Zhang, Zhimin Zhang
2020 arXiv   pre-print
In this paper, we turn our focus to 3 dimensions and construct an H(curl^2)-conforming tetrahedral finite element.  ...  In [24], we proposed H(curl^2)-conforming elements on both a triangle and a rectangle. This family of elements provides a brand new method to solve the quad-curl problem in 2 dimensions.  ...  In [23] , the authors and another collaborator constructed curl-curl-conforming or H(curl 2 )-conforming elements in 2 dimensions (2D) to solve the quad-curl problem.  ... 
arXiv:2007.10421v2 fatcat:7lmzlnzjc5dthnosrvbqvpkufu

Finite Element Methods for Maxwell's Transmission Eigenvalues

Peter Monk, Jiguang Sun
2012 SIAM Journal on Scientific Computing  
In this paper, we propose two finite element methods to compute a few lowest Maxwell's transmission eigenvalues which are of interest in applications.  ...  In *  ...  In Section 4, we propose two finite element methods, i.e., the curl conforming finite element method and the mixed finite element method.  ... 
doi:10.1137/110839990 fatcat:d3roo5575ncavlaf74xuelmzze

Bernstein–Bézier bases for tetrahedral finite elements

Mark Ainsworth, Guosheng Fu
2018 Computer Methods in Applied Mechanics and Engineering  
We present a new set of basis functions for H(curl)-conforming, H(div)-conforming, and L2 -conforming finite elements of arbitrary order on a tetrahedron.  ...  The basis functions are expressed in terms of Bernstein polynomials and augment the natural H1 -conforming Bernstein basis.  ...  In two dimensions, see (9) and (12) , the non-gradients consist of 2D lowest-order edge elements, and H(curl) face bubbles.  ... 
doi:10.1016/j.cma.2018.05.034 fatcat:tlqmh3q4tvcb3m73u6ktfr7wi4

H(curl$^2$)-Conforming Finite Elements in 2 Dimensions and Applications to the Quad-Curl Problem

Qian Zhang, Lixiu Wang, Zhimin Zhang
2019 SIAM Journal on Scientific Computing  
In this paper, we construct some H(curl 2 )-conforming finite elements on a rectangle (a parallelogram) and a triangle.  ...  Then we apply our new elements to construct a finite element space for discretizing the quad-curl problem.  ...  We would like to thank Professor Jiguang Sun for drawing our attention to the quad-curl problem and for his valuable comments and suggestions.  ... 
doi:10.1137/18m1199988 fatcat:fgomzfvg6fcpjasd5egtmgvski
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