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This study focused on the superconvergence of the finite element method for the fivedimensional Poisson equation in the W1,∞-seminorm. Specifically, we investigated the block finite element method, which is a tensor-product finite element approach applied to regular rectangular partitions of the domain. First, we introduced the finite element scheme for the equation and discussed various functions related to it, along with their properties. Next, we proposed a weight function and established its important properties, which play a crucial role in the theoretical analysis. By utilizing the properties of the weight function and employing weighted-norm analysis techniques, we derived an optimal order estimate in the W2,1-seminorm for the discrete derivative Green's function (DDGF). Furthermore, we provided an interpolation fundamental estimate of the second type, also known as the weak estimate of the second type, for the block finite element. This weak estimate is based on a five-dimensional interpolation operator of the projection type. Finally, by combining the derived W2,1-seminorm estimate for the DDGF and the weak estimate for the block finite element, we obtained a superconvergence estimate for the block finite element approximation in the pointwise sense of the W1,∞-seminorm. The correctness of the theoretical results was demonstrated through a numerical example. [ABSTRACT FROM AUTHOR]

In our previous article, we introduced the definition of the 3D Green's function, and gave some estimates for this function. In this article, we will give the finite element approximation to the 3D Green's function. Moreover, some error estimates between 3D Green's function and its finite element approximation are derived, which will be used to the local superconvergence analysis. [ABSTRACT FROM AUTHOR]

This article will first introduce the definition of the Green's function of 3D elliptic equations, which plays important roles in local superconvergence estimates for the finite element approximation. Then, using the weighted-norm methods, we derive some estimates for the 3D Green's function. [ABSTRACT FROM AUTHOR]

In this article we first introduce definitions of the regularized Green's function, the discrete Green's function, the discrete δ function, and the L2-projection operator in six dimensions. Then the W2,1-seminorm estimates for the regularized Green's function and the discrete Green's function are derived. Finally, pointwise superconvergence of the displacement of the six-dimensional finite element is obtained. [ABSTRACT FROM AUTHOR]

This article will discuss estimates for discrete derivative Green's function for elliptic equations in dimensions seven and up. First, the definitions of some terms are given. Then the estimates for the regularized derivative Green's function are derived. Finally, using the triangular inequality, we obtain the estimates for discrete derivative Green's function. The results of the article play important roles in the research of superconvergence of finite element methods. [ABSTRACT FROM AUTHOR]

In this article we first introduce definitions of the regularized derivative Green's function, the discrete derivative Green's function, the discrete derivative δ function, and the L²-projection operator in four dimensions. Then the W2, 1,-seminorm estimates for the regularized derivative Green's function and the discrete derivative Green's function are derived. Finally, we show the applications of the W2, 1-seminorm estimate for the discrete derivative Green's function in finite element (FE) superconvergence. [ABSTRACT FROM AUTHOR]

In this article we first introduce definitions of the regularized Green's function, the discrete Green's function, the discrete δ function, and the L² projection operator in five dimensions. Then the W2,1-norm estimate for the regularized Green's function is derived. Finally, we obtain the W2,1-seminorm estimate for the discrete Green's function [ABSTRACT FROM AUTHOR]

We consider the finite element approximation to the solution of a self-adjoint, second-order elliptic boundary value problem in three dimensions over a fully uniform mesh of piecewise linear tetrahedral elements. First, the supercloseness of the gradients between the piecewise linear finite element solution uh and the linear interpolation uI is derived by using a weak estimate and an estimate of the discrete derivative Green's function. We then analyze a superconvergence patch recovery scheme for the gradient of the finite element solution, showing that the recovered gradient of uh is superconvergent to the gradient of the true solution u. [ABSTRACT FROM AUTHOR]

For a second order variable coefficient elliptic boundary value problem in 3D, we derive the weak estimate of the first type for tensor-product quadratic pentahedral finite elements. Combined the estimate for the discrete derivative Green's function, we show that the derivatives of the finite element approximation and the corresponding interpolant are superclose in the sense of the L∞-norm. [ABSTRACT FROM AUTHOR]

For the Poisson equation in three dimensions, using the properties of the interpolation operator of projection type, we derive weak estimates for cubic block finite elements of intermediate families (CBFEIF). In addition, estimates for the discrete Green's functions are given. Finally, we prove that derivatives and function values of the finite element approximation and the corresponding interpolant of projection type are superclose in the pointwise sense of the L∞-norm. [ABSTRACT FROM AUTHOR]

In this paper, definitions of the regularized derivative Green's function ∂z,ℓDue to image rights restrictions, multiple line equation(s) cannot be graphically displayed. and the discrete derivative Green's function ∂z,ℓDue to image rights restrictions, multiple line equation(s) cannot be graphically displayed. in four dimensions are first given. With complicated arguments, the W1,1-seminorm estimate for ∂z,ℓDue to image rights restrictions, multiple line equation(s) cannot be graphically displayed. is then derived. Finally, applying the triangle inequality and the error estimate of the finite element approximation results in the ∂z,ℓDue to image rights restrictions, multiple line equation(s) cannot be graphically displayed. W1,1-seminorm estimate with order O(| ln h| 5/4) for ∂z,ℓDue to image rights restrictions, multiple line equation(s) cannot be graphically displayed. [ABSTRACT FROM AUTHOR]

In this paper, we first introduce the definitions of the regularized derivative Green's function Due to image rights restrictions, multiple line equation(s) cannot be graphically displayed. and the discrete derivative Green's function Due to image rights restrictions, multiple line equation(s) cannot be graphically displayed. for the Poisson equation in five dimensions. Then the W1,1-seminorm estimate for Due to image rights restrictions, multiple line equation(s) cannot be graphically displayed. is derived. Finally, applying the triangle inequality and the error estimate of the finite element approximation, we obtain the W1,1-seminorm estimate with order O(| ln h|7/5) for Due to image rights restrictions, multiple line equation(s) cannot be graphically displayed. [ABSTRACT FROM AUTHOR]