In this paper, we introduce a new approach to solving the porous medium equation using a moving mesh finite element method that leverages the Onsager variational principle as an approximation tool. Both the continuous and discrete problems are formulated based on the Onsager principle. The energy dissipation structure is maintained in the semi-discrete and fully implicit discrete schemes. We also develop a fully decoupled explicit scheme by which only a few linear equations are solved sequentially in each time step. The numerical schemes exhibit an optimal convergence rate when the initial mesh is appropriately selected to ensure accurate approximation of the initial data. Furthermore, the method naturally captures the waiting time phenomena without requiring any manual intervention.

翻译：本文提出了一种采用移动网格有限元方法求解多孔介质方程的新途径，该方法将昂萨格变分原理作为近似工具。连续问题与离散问题均基于昂萨格原理进行构建。半离散格式与全隐式离散格式均保持了能量耗散结构。我们还发展了一种完全解耦的显式格式，该格式在每个时间步仅需顺序求解少量线性方程。当初始网格适当选取以精确逼近初始数据时，数值格式展现出最优收敛速率。此外，该方法无需任何人工干预即可自然地捕捉等待时间现象。