A superconvergent nonconforming quadrilateral spline element for biharmonic equation using the B-net method

Abstract

In this paper, we construct a new nonconforming quadrilateral element with 12 degrees of freedom to solve the biharmonic problems. ${\mathscr {T}}_{h}$ is a triangulated quadrangulation of the domain $\varOmega$. For a quadrilateral element $Q_{T}$, the finite element space, which contains ${\mathbb {P}}_{3}(Q_{T})$, is a subspace of the bivariate spline space ${\mathbf {S}}_{3}^{1}(Q_{T})$. The degrees of freedom are chosen as the four point values, the four edge integrals of the shape functions and the edge integrals of their normal derivatives such that the weak continuity between elements can be satisfied. Accordingly, we explicitly establish 12 spline interpolation bases in the B-net form. Error estimates are given with optimal convergence order in both discrete $H^{2}$ and $H^{1}$ seminorms. The proposed element NCQS12 can get the superconvergence results with theoretical proof when ${\mathscr {T}}_{h}$ is an uniform parallelogram mesh. Some degenerate meshes are considered subsequently. Finally, we do some numerical experiments to verify the theoretical analysis. Numerical results show that the proposed element performs well over the asymptotically regular parallelogram meshes, which is same as over the uniform parallelogram meshes.