\(C^{0}\)-nonconforming tetrahedral and cuboid elements for the three-dimensional fourth-order elliptic problem (Q1950442)

The authors prove an abstract convergence theorem that gives a theoretical frame to construct \(C^{0}\)-nonconforming elements for a fourth-order elliptic problem. The shape function space is divided into two subspaces by using bubble functions. One subspace is responsible for the \(C^{0}\)-continuity of the shape functions and for getting the approximation error. The other one (containing the bubble functions) is responsible for the continuity in the mean of the normal derivatives of the shape functions across the elements and for getting the consistence error. The authors construct a tetrahedral \(C^{0}\)-nonconforming element and two cuboid \(C^{0}\)-nonconforming elements for the three-dimensional fourth-order problem. It is also proved that one element is of first order convergence and other two are of second order convergence. The authors claim that for the first time, a second-order convergent nonconforming element for the fourth-order elliptic problem is constructed in this paper.