Novel Raviart–Thomas Basis Functions on Anisotropic Finite Elements

Fleurianne Bertrand

doi:10.1515/cmam-2022-0235

Published by De Gruyter September 19, 2023

Novel Raviart–Thomas Basis Functions on Anisotropic Finite Elements

Fleurianne Bertrand

From the journal Computational Methods in Applied Mathematics

https://doi.org/10.1515/cmam-2022-0235

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Abstract

Recently, H ⁢ ( div ) -conforming finite element families were proven to be successful on anisotropic meshes, with the help of suitable interpolation error estimates. In order to ensure corresponding large-scale computation, this contribution provides novel Raviart–Thomas basis functions, robust regarding the anisotropy of a given triangulation. This new set of basis functions on simplices uses a hierarchical approach, and the orientation of the basis functions is inherited from the lowest-order case. In the higher-order case, the new basis functions can be written as a combination of the lowest-order Raviart–Thomas elements and higher-order Lagrange-elements. This ensures robustness regarding the mesh anisotropy and assembling strategies as demonstrated in the numerical experiments.

Keywords: Anisotropic Elements; Raviart–Thomas; Mixed Finite Elements

MSC 2010: 65N30

Dedicated to Thomas Apel on the occasion of his 60th birthday.

A Basis Functions Commonly Used in Literature

This section lists some of the basis functions used in the literature. For the lowest-order case, only the choices

ψ σ i , 0 ⁢ ( T ) = { 1 2 ⁢ | T | ⁢ ( x − ν i ) if ⁢ n = n E T , i , − 1 2 ⁢ | T | ⁢ ( x − ν i ) if ⁢ n = − n E T , i , and ψ σ ̃ i , 0 ⁢ ( T ) = | E T , i | ⁢ ψ σ i , 0 ⁢ ( T )

from (4.1) commonly occur:

use of ψ σ ̃ i , 0 ⁢ ( T ) in [8, 29, 27, 20],
use of ψ σ i , 0 ⁢ ( T ) in [30, 16, 17].

For the higher-order case, different combinations occur.

	Edge-based basis functions	Volume-based basis functions
[29]	{ v E , p } E ∈ E ⁢ ( T ) , p ∈ Ξ E with Ξ E defined in (7.1)	{ v T , p , d } T ∈ T , p ∈ Θ , d ∈ I d from (6.1)
[20]	{ v E , p } E ∈ E ⁢ ( T ) , p ∈ Ξ E with Ξ E = L k ⁢ ( E )	{ v T , p , d } T ∈ T , p ∈ Θ , d ∈ I d from (6.1)
[27]	Corresponding to (3.2a)	Corresponding to (3.2b)
[30]	{ v E , p } E ∈ E ⁢ ( T ) , p ∈ Ξ E with Ξ E = L k ⁢ ( E )	{ v T , p , d } T ∈ T , p ∈ Θ , d ∈ I d from (6.1)

Acknowledgements

At the birthday dinner honoring Thomas Apel, his remarkable contributions to the field of finite element were recognized and lauded. Even though I cannot compete with the esteemed experts present there in terms of collaborations and joint publications, let me emphasize that Thomas Apel embodies humility and kindness, and this also towards less experienced scientists. He consistently maintains an open-minded approach and demonstrates genuine interest in the numerous inexperienced students he encounters, in particular at the Finite Element Symposium. Thomas, let me express my heartfelt gratitude to you for creating such an environment where I felt encouraged to learn from my errors and grow, supported and appreciated.

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Received: 2022-11-19

Revised: 2023-07-06

Accepted: 2023-07-07

Published Online: 2023-09-19

Published in Print: 2023-10-01

Novel Raviart–Thomas Basis Functions on Anisotropic Finite Elements

Abstract

A Basis Functions Commonly Used in Literature

Acknowledgements

References

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