Abstract The purpose of this paper is to prove a priori error estimates for the completely discretized problem of the dual mixed method for the non-stationary heat diffusion equation in a polygonal domain of R 2 . Due to the geometric singularities of the domain, the exact solution is not regular in the context of classical Sobolev spaces. Instead, one must use weighted Sobolev spaces in our analysis. To obtain optimal convergence rates of the discrete solutions, it is necessary to refine adequately the considered meshings near the reentrant corners of the polygonal domain. In a previous work, using the Raviart–Thomas vectorfields of degree 0 for the discretization of the heat flux density vector, we have obtained a priori error estimates of order 1 for the semi-discrete solutions of this problem. In our actual paper, we complete the discretization of the problem in time by using Euler’s implicit scheme, and we obtain optimal error estimates of order 1, in time and space. Numerical results are given to illustrate this improvement of the convergence orders.

中文翻译：

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多边形域热扩散方程对偶混合法的完全离散化

摘要 本文的目的是证明R 2 多边形域中非平稳热扩散方程的对偶混合法完全离散化问题的先验误差估计。由于域的几何奇点，精确解在经典 Sobolev 空间的上下文中是不规则的。相反，我们必须在我们的分析中使用加权 Sobolev 空间。为了获得离散解的最佳收敛速度，有必要在多边形域的可重入角附近充分细化所考虑的网格。在之前的工作中，使用 0 次的 Raviart-Thomas 矢量场对热通量密度矢量进行离散化，我们已经获得了该问题半离散解的 1 阶先验误差估计。在我们的实际论文中，我们利用欧拉隐式格式及时完成了问题的离散化，得到了时间和空间上的1阶最优误差估计。给出了数值结果来说明收敛阶数的这种改进。