\frac{1}{\lambda^d(t)} 
            \left( 
            1 - \left( \frac{r}{r_0 \lambda(t)} \right)^2 
            \right)^{\frac{1}{m}} , \quad & \text{for} \quad |r| \leq r_0 \lambda(t)  \\
                                0 , \quad & \text{for} \quad |r| > r_0 \lambda(t)  
        \end{cases}
        \label{BP-soln}$$ where $$r = | {\boldsymbol{x}} | , \quad 
        \lambda(t) = \left(\frac{t}{t_0} \right)^{\frac{1}{2+d m}} ,
        \quad 
        t_0 = \frac{r_0^2 m}{2 (2 + d m)} ,$$ and $r_0>0$ is a given parameter. It is radially symmetric, self-similar, and compact-supported for any finite time. Moreover, $u^m$ is Lipschitz continuous in ${\boldsymbol{x}}$ and $t$ and $\nabla (u^m)$ is bounded in the support of $u(\cdot, t)$, $\text{supp}(u(\cdot, t))$. Furthermore, the solution is Hölder continuous. The slope of the solution at the free boundary is finite for $m = 1$ and becomes infinite when $m>1$, which causes challenges for the numerical solution of PME. These regularity properties also hold for general compactly supported solutions of (\[PME-1\]), and their free boundaries can be shown to be at least Lipschitz continuous in both space and time; e.g., see [@Aronson1969; @Caffarelli1980; @Daskalopoulos1998a; @Shmarev2005].

The moving mesh finite element method {#SEC:mmfem}
=====================================

In this section we describe the adaptive moving mesh finite element approximation of IBVP . To begin with, we note that there are roughly two approaches for 