LAUSR

In this paper, we propose a novel difference finite element (DFE) method based on the P 1 -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 h n ${u^{n}_{h}}$ by the second-order backward difference formula in the temporal t -direction, the center difference scheme in the spatial z -direction, and the P 1 -element on a almost-uniform mesh J h in the spatial ( x , y )-direction. Next, the H 1 -stability of u h n ${u_{h}^{n}}$ and the second-order H 1 -convergence of the interpolation post-processing function on u h n ${u_{h}^{n}}$ with respect to u ( t n ) are provided. Finally, numerical tests are presented to show the second-order H 1 -convergence results of the proposed DFE method for the heat equation in a 3D spatial domain.