Abstract: For Dirichlet boundary value problems of Poisson equations, an interpolation operator of projection type in 4D was established. Then by using this operator, weak estimates for tensor-product quadratic rectangular finite elements over regular partitions of a domain were obtained. Based on the obtained results and the estimates for the four-dimensional discrete Green's function, some highly accuracy results of the maximum-norm superapproximations of finite elements were derived.

Key words: elliptic boundary value problem, interpolation operator of projection type in 4D, rectangular finite element, weak estimate, superapproximation