Assume that $u\left(x\right)$ satisfies the problem $\mathcal{L}u\left(x\right)\equiv -\frac{\partial }{\partial x_i}\left(a_{ij}\frac{\partial u}{\partial x_j}\right)=f\left(x\right),\forall x\in \Omega ,u\left(x\right)=0,\forall x\in \partial \Omega$. In this article, using interpolation postprocessing technique, we will investigate the local ultraconvergence of the primal variable and the derivative of finite element approximation of $u\left(x\right)$ using piecewise polynomials of degrees bi-$k(k\ge 3)$ over a rectangular partition. Assume that $k\ge 3$ is odd and $x_0$ is an interior vertex satisfying $\rho (x_0,\partial \Omega )\ge c$. Using the new interpolation postprocessing formula presented in this study, we show that the primal variable and the derivative of the post-processed finite element solution using piecewise of degrees bi-$k(k\ge 3)$ at $x_0$ converge to the primal variable and the derivative of the exact solution with order $\mathcal{O}\left(h^{k+3}|lnh|\right)$ under suitable regularity and mesh conditions, respectively. Finally, we use numerical experiments to illustrate our theoretical findings.

假使，假设 $ü（X）$ 满足问题 $\mathcal{大号}ü（X）\equiv -\frac{\partial }{\partial X_{\mathrm{一世}}}（{\mathrm{一种}}_{\mathrm{一世}Ĵ}\frac{\partial ü}{\partial X_{Ĵ}}）=F（X），\forall X\in \Omega ，ü（X）=0，\forall X\in \partial \Omega$。在本文中，我们将使用插值后处理技术来研究原始变量的局部超收敛性和有限元逼近的导数。$ü（X）$ 使用度数为bi-的分段多项式$ķ（ķ\ge 3）$在矩形分区上。假使，假设$ķ\ge 3$ 是奇数 $X_0$ 是一个令人满意的内部顶点 $\rho （X_0，\partial \Omega ）\ge C$。使用本研究中提出的新的插值后处理公式，我们证明了使用双度数分段的后处理有限元解的原始变量和导数$ķ（ķ\ge 3）$ 在 $X_0$ 收敛到原始变量和阶次精确解的导数 $\mathcal{Ø}（H^{ķ+3}|lnH|）$分别在适当的规则性和网格条件下。最后，我们使用数值实验来说明我们的理论发现。