Year: 2023
Author: Jing Chen, Zhaojie Zhou, Huanzhen Chen, Hong Wang
Journal of Computational Mathematics, Vol. 41 (2023), Iss. 5 : pp. 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.
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Journal Article Details
Publisher Name: Global Science Press
Language: English
DOI: https://doi.org/10.4208/jcm.2112-m2021-0204
Journal of Computational Mathematics, Vol. 41 (2023), Iss. 5 : pp. 817–840
Published online: 2023-01
AMS Subject Headings:
Copyright: COPYRIGHT: © Global Science Press
Pages: 24
Keywords: New finite element space Expanded mixed finite element Minimum degrees of freedom The inf-sup condition Solvability Optimal convergence.