H¹‐superconvergence of finite difference method based on Q₁‐element on quasi‐uniform mesh for the 3D Poisson equation

Abstract

In this paper, the finite difference (FD) method is considered for the 3D Poisson equation by using the Q1‐element on a quasi‐uniform mesh. First, under the regularity assumption of u∈H3Ω∩H01Ω, the H1‐superconvergence of the FD solution uh based on the Q1‐element to the first‐order interpolation function Ih1u is obtained. Next, the H1‐superconvergence of the second‐order interpolation postprocessing function I2h2uh based on the FD solution uh to u is provided. Finally, numerical tests are presented to show the H1‐superconvergence result of the FD postprocessing function I2h2uh to u if u∈H3Ω∩H01Ω.

Stability and Convergence of the Integral-Averaged Interpolation Operator Based on $Q_1$-Element in $\mathbb{R}^n$

“…The interpolation theory of functions is an important part of the theoretical analysis of the finite element method. It is used to estimate the value of the function at other points from the values at known data points [1][2][3][6][7][8]. In practical applications, the selection of an appropriate interpolation operator is crucial to ensure computational accuracy and stability in the numerical methods for partial differential equations.…”