This paper presents a fitted space-time finite element method for solving a parabolic advection-diffusion problem with a nonstationary interface. The jumping diffusion coefficient gives rise to the discontinuity of the spatial gradient of solution across the interface. We use the Banach-Necas-Babuska theorem to show the well-posedness of the continuous variational problem. A fully discrete finite-element based scheme is analyzed using the Galerkin method and unstructured fitted meshes. An optimal error estimate is established in a discrete energy norm under appropriate globally low but locally high regularity conditions. Some numerical results corroborate our theoretical results.

翻译：本文提出了一种拟合时空有限元方法，用于求解具有非平稳界面的抛物型对流扩散问题。跳跃的扩散系数导致解的空间梯度在界面处不连续。我们利用Banach-Necas-Babuska定理证明了连续变分问题的适定性。采用Galerkin方法和非结构拟合网格对全离散有限元格式进行了分析。在适当的全局低、局部高正则性条件下，于离散能量范数中建立了最优误差估计。数值结果验证了理论分析的正确性。