A New Finite Element Space for Expanded Mixed Finite Element Method

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.