A New Finite Element Space for Expanded Mixed Finite Element Method (Journal of Computational Mathematics, 2023)
In this article, we propose a new finite element space $Λ_h$ for the expanded mixedfinite 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 classicalcounterpart. Furthermore, the new $Λ_h$ coupled with the Raviart-Thomas space satisfiesthe inf-sup condition, which is crucial to the computation of mixed methods for its closerelation to the behavior of the smallest nonzero eigenvalue of the stiff matrix, and thusthe existence, uniqueness and optimal approximate capability of the EMFEM solution areproved 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 withoutthe inf-sup condition. Numerical experiments are conducted to confirm these theoreticalfindings.