In this paper, we propose a novel difference finite element (DFE) method based on the P₁-element for the 3D heat equation on a 3D bounded domain. One of the novel ideas of this paper is to use the second-order backward difference formula (BDF) combining DFE method to overcome the computational complexity of conventional finite element (FE) method for the high-dimensional parabolic problem. First, we design a fully discrete difference FE solution \({u^{n}_{h}}\) by the second-order backward difference formula in the temporal t-direction, the center difference scheme in the spatial z-direction, and the P₁-element on a almost-uniform mesh J_h in the spatial (x, y)-direction. Next, the H¹-stability of \({u_{h}^{n}}\) and the second-order H¹-convergence of the interpolation post-processing function on \({u_{h}^{n}}\) with respect to u(t_n) are provided. Finally, numerical tests are presented to show the second-order H¹-convergence results of the proposed DFE method for the heat equation in a 3D spatial domain.

在本文中，我们针对3D有界域上的3D热方程提出了一种基于P ₁元素的新型差分有限元（DFE）方法。本文的新颖思想之一是结合DFE方法使用二阶后向差分公式（BDF）来克服传统有限元（FE）方法在高维抛物线问题上的计算复杂性。首先，我们通过时间t方向上的二阶后向差分公式，空间z方向上的中心差分方案设计完全离散的差分FE解\（{u ^ {n} _ {h}} \），以及几乎均匀的网格J _h上的P ₁元素在空间（x，y）方向上。接着，ħ ¹的-稳定性\（{U_ {H} ^ {N}} \）和二阶ħ ¹的后处理内插函数的上-convergence \（{U_ {H} ^ {N} } \）关于u（t _n）被提供。最后，数值试验表明了所提出的DFE方法在3D空间域中的热方程的二阶H ¹收敛结果。