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Volume 41, Issue 5
A New Finite Element Space for Expanded Mixed Finite Element Method

Jing Chen, Zhaojie Zhou, Huanzhen Chen & Hong Wang

DOI: 10.4208/jcm.2112-m2021-0204

J. Comp. Math., 41 (2023), pp. 817-840.

Published online: 2023-05

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  • Abstract

In this article, we propose a new finite element space $Λ_h$ for the expanded mixed finite element method (EMFEM) for second-order elliptic problems to guarantee its computing capability and reduce the computation cost. The new finite element space $Λ_h$ is designed in such a way that the strong requirement $V_h ⊂ Λ_h$ in [9] is weakened to $\{v_h ∈ V_h; {\rm div} v_h = 0\} ⊂ Λ_h$ so that it needs fewer degrees of freedom than its classical counterpart. Furthermore, the new $Λ_h$ coupled with the Raviart-Thomas space satisfies the inf-sup condition, which is crucial to the computation of mixed methods for its close relation to the behavior of the smallest nonzero eigenvalue of the stiff matrix, and thus the existence, uniqueness and optimal approximate capability of the EMFEM solution are proved for rectangular partitions in $\mathbb{R}^d$, $d = 2, 3$ and for triangular partitions in $\mathbb{R}^2.$ Also, the solvability of the EMFEM for triangular partition in $\mathbb{R}^3$ can be directly proved without the inf-sup condition. Numerical experiments are conducted to confirm these theoretical findings.

  • Keywords

New finite element space, Expanded mixed finite element, Minimum degrees of freedom, The inf-sup condition, Solvability, Optimal convergence.

  • AMS Subject Headings

65N30

  • Copyright

COPYRIGHT: © Global Science Press

  • Email address

614005@sdnu.edu.cn (Jing Chen)

zhouzhaojie@sdnu.edu.cn (Zhaojie Zhou)

chhzh@sdnu.edu.cn (Huanzhen Chen)

hwang@math.sc.edu (Hong Wang)

  • BibTex
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  • TXT
@Article{JCM-41-817, author = {Chen , JingZhou , ZhaojieChen , Huanzhen and Wang , Hong}, title = {A New Finite Element Space for Expanded Mixed Finite Element Method}, journal = {Journal of Computational Mathematics}, year = {2023}, volume = {41}, number = {5}, pages = {817--840}, abstract = {

In this article, we propose a new finite element space $Λ_h$ for the expanded mixed finite element method (EMFEM) for second-order elliptic problems to guarantee its computing capability and reduce the computation cost. The new finite element space $Λ_h$ is designed in such a way that the strong requirement $V_h ⊂ Λ_h$ in [9] is weakened to $\{v_h ∈ V_h; {\rm div} v_h = 0\} ⊂ Λ_h$ so that it needs fewer degrees of freedom than its classical counterpart. Furthermore, the new $Λ_h$ coupled with the Raviart-Thomas space satisfies the inf-sup condition, which is crucial to the computation of mixed methods for its close relation to the behavior of the smallest nonzero eigenvalue of the stiff matrix, and thus the existence, uniqueness and optimal approximate capability of the EMFEM solution are proved for rectangular partitions in $\mathbb{R}^d$, $d = 2, 3$ and for triangular partitions in $\mathbb{R}^2.$ Also, the solvability of the EMFEM for triangular partition in $\mathbb{R}^3$ can be directly proved without the inf-sup condition. Numerical experiments are conducted to confirm these theoretical findings.

}, issn = {1991-7139}, doi = {https://doi.org/10.4208/jcm.2112-m2021-0204}, url = {http://global-sci.org/intro/article_detail/jcm/21675.html} }
TY - JOUR T1 - A New Finite Element Space for Expanded Mixed Finite Element Method AU - Chen , Jing AU - Zhou , Zhaojie AU - Chen , Huanzhen AU - Wang , Hong JO - Journal of Computational Mathematics VL - 5 SP - 817 EP - 840 PY - 2023 DA - 2023/05 SN - 41 DO - http://doi.org/10.4208/jcm.2112-m2021-0204 UR - https://global-sci.org/intro/article_detail/jcm/21675.html KW - New finite element space, Expanded mixed finite element, Minimum degrees of freedom, The inf-sup condition, Solvability, Optimal convergence. AB -

In this article, we propose a new finite element space $Λ_h$ for the expanded mixed finite element method (EMFEM) for second-order elliptic problems to guarantee its computing capability and reduce the computation cost. The new finite element space $Λ_h$ is designed in such a way that the strong requirement $V_h ⊂ Λ_h$ in [9] is weakened to $\{v_h ∈ V_h; {\rm div} v_h = 0\} ⊂ Λ_h$ so that it needs fewer degrees of freedom than its classical counterpart. Furthermore, the new $Λ_h$ coupled with the Raviart-Thomas space satisfies the inf-sup condition, which is crucial to the computation of mixed methods for its close relation to the behavior of the smallest nonzero eigenvalue of the stiff matrix, and thus the existence, uniqueness and optimal approximate capability of the EMFEM solution are proved for rectangular partitions in $\mathbb{R}^d$, $d = 2, 3$ and for triangular partitions in $\mathbb{R}^2.$ Also, the solvability of the EMFEM for triangular partition in $\mathbb{R}^3$ can be directly proved without the inf-sup condition. Numerical experiments are conducted to confirm these theoretical findings.

Chen , JingZhou , ZhaojieChen , Huanzhen and Wang , Hong. (2023). A New Finite Element Space for Expanded Mixed Finite Element Method. Journal of Computational Mathematics. 41 (5). 817-840. doi:10.4208/jcm.2112-m2021-0204
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