governed by the ${\boldsymbol{x}}$-formulation will stay nonsingular when it is nonsingular initially. Although such a theoretical result is not available for the ${\boldsymbol{\xi}}$-formulation, our limited numerical experience shows that the ${\boldsymbol{\xi}}$-formulation also produces nonsingular meshes.

Numerical Results for PME {#SEC:PME-numerics}
=========================

In this section we present numerical results obtained with the moving mesh finite element method described in the previous section for a number of PME examples. They include the Barenblatt-Pattle solution and the generalizations of several one-dimensional examples studied by Zhang and Wu [@Zhang-2009]. These examples are selected to demonstrate the accuracy of our method as well as its ability to deal with solutions having complex support and the waiting-time phenomenon. For the cases having an exact solution, the error in the computed solution will be measured in the (global) $L^2$ norm, i.e., $$\|e_h\|_{L^2(t_0,T; L^2(\Omega))} = \left ( \int_{t_0}^{T} \int_{\Omega} e_h^2({\boldsymbol{x}}, t) d {\boldsymbol{x}}  d t \right )^{\frac{1}{2}}.$$ We choose this norm because various error estimates are obtained in this norm, e.g., see (\[ebmeyer-1\]). (An exception is Fig. \[fig:pme4-adaptivity-L1\] where the convergence history in $L^1$ norm is plotted for comparison purpose.) In our computation, we use $\tau = 10^{-4}$ (for the mesh movement), the maximal allowed time step size 