Superconvergence analysis of cubic block finite elements of intermediate families (Q2912685)

The paper studies the solution of Poisson's equation in 3D with cubic finite elements of an intermediate family of type II, i.e., the function is in each 3D block a cubic polynomial belonging to the complete polynomial space of degree 5. The dimension of this space is 38 compared to the dimension 64 of the cubic tensor product polynomial space. The author establishes weak estimates for the projection operator to the finite element space on a regular family of rectangular partitions. Then estimates for discrete Green's functions lead to superconvergence results, for example of the order \(h^5 | \ln h|^{2/3}\) in \(L^{\infty}\) for sufficiently smooth solutions.