A staggered mixed method for the biharmonic problem based on the first-order system

Abstract

In this paper a staggered mixed method based on the first-order system is proposed and analysed. The proposed method uses piecewise polynomial spaces enjoying partial continuity properties and hinges on a careful balancing of the involved finite element spaces. It supports polygonal meshes of arbitrary shapes and is free of stabilization. All the variables converge optimally with respect to the polynomial order as demonstrated by the rigorous convergence analysis. In particular, it is shown that the approximations of $u$ and $\boldsymbol{p}:=\nabla u$ superconverge to the suitably defined projections, and it is noteworthy that the approximation of $u$ superconverges to the projection in $L^{2}$-error of order $k+3$ up to the data oscillation term when polynomials of degree $k-1$ are used for the approximation of $u$. Taking advantage of the superconvergence we are able to define the local postprocessing approximations for $u$ and $\boldsymbol{p}$, respectively. The convergence error estimates for the postprocessing approximations are also proved. Several numerical experiments are presented to confirm the proposed theories.