Baines et al. [@BHJ05a] (conservation of fraction mass), and Budd and Williams [@BW06] (parabolic [M]{}onge-[A]{}mpère equation). The interested reader is also referred to the books/review articles [@Bai94a; @Baines-2011; @Shashkov-2015; @BHR09; @HR11; @Tan05] and references therein. The outline of the paper is as follows. Some properties of PME that are relevant to the numerical simulation are described in Section \[SEC:PME-theory\]. Section \[SEC:mmfem\] is devoted to the description of the moving mesh finite element method, including the linear finite element discretion of PME on a moving mesh and the generation of an adaptive moving mesh using the MMPDE approach. Numerical examples are presented in Sections \[SEC:PME-numerics\] and \[SEC:PME-numerics-2\] for PME and PME with variable exponents and absorption, respectively. Finally, Section \[SEC:conclusion\] contains the conclusions and further comments.

Properties of the porous medium equation {#SEC:PME-theory}
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Before we describe the finite element approximation of IBVP (\[PME-1\]), it is instructive to recall some of its properties that are relevant to numerical simulation. First of all, it is known (e.g., see V[á]{}zquez [@Vazquez2007]) that when $u_0^{m+2} \in L^{1}(\Omega)$, IBVP has a weak solution $u$ satisfying $u(\cdot, t)^{m+2} \in L^{1}(\Omega)$ for any $t  \in (t_0, T]$ and $u^{m+1} \in L^{2}(0,T; H^1_0(\Omega))$. Moreover, a nonnegative weak solution 