In this work, we present the construction of two distinct finite element approaches to solve the Porous Medium Equation (PME). In the first approach, we transform the PME to a log-density variable formulation and construct a continuous Galerkin method. In the second approach, we introduce additional potential and velocity variables to rewrite the PME into a system of equations, for which we construct a mixed finite element method. Both approaches are first-order accurate, mass conserving, and proved to be unconditionally energy stable for their respective energies. The mixed approach is shown to preserve positivity under a CFL condition, while a much stronger property of unconditional bound preservation is proved for the log-density approach. A novel feature of our schemes is that they can handle compactly supported initial data without the need for any perturbation techniques. Furthermore, the log-density method can handle unstructured grids in any number of dimensions, while the mixed method can handle unstructured grids in two dimensions. We present results from several numerical experiments to demonstrate these properties.

翻译：本文提出了两种求解多孔介质方程(PME)的不同有限元方法。第一种方法将PME转化为对数密度变量形式，并构造了连续Galerkin方法。第二种方法引入额外的势函数和速度变量，将PME改写为方程组，并为其构造了混合有限元方法。两种方法均具有一阶精度、质量守恒特性，并已被证明对于各自对应的能量具有无条件能量稳定性。混合方法在CFL条件下可保持正性，而对数密度方法则被证明具有更强的无条件有界保持性质。本方案的一个新颖特性在于能够处理紧支撑初始数据而无需任何扰动技术。此外，对数密度方法可处理任意维度的非结构化网格，而混合方法仅能处理二维非结构化网格。我们通过多个数值实验的结果展示了这些特性。