Computational Bases for $$\mathcal {S}_{r}\Lambda ^{1}(\mathbb {R}^{2})$$ S r Λ 1 ( R 2 ) and Their Application in Mixed Finite Element Method

Abstract

In this article, computational bases for finite element spaces $$\mathcal {S}_{2}\Lambda ^{0}(\mathcal {T}_{h})$$ and $$\mathcal {S}_{r}\Lambda ^{1}(\mathcal {T}_{h})$$ in each step of the h-adaptive method are derived. We also implement a mixed method for the Hodge Laplacian equation. In discretization of the mixed method, the pair which consists of the serendipity elements and the rectangular Brezzi–Douglas–Marini (BDM) elements are used. The corresponding saddle point matrix, in each step of the h-adaptive method, is $$\begin{aligned} \mathcal {A}=\begin{pmatrix} A&{}B^{T}\\ -B&{} C \end{pmatrix}, \end{aligned}$$ where $$ C\ne 0 $$ and B is rank deficient. The modified generalized shift-splitting (MGSS) preconditioner for solving this saddle point matrix is considered. The major advantage of our approach is that the MGSS preconditioner can easily be implemented. Numerical results show the effectiveness of the proposed iteration method and the good behavior of corresponding splitting preconditioner.