that the linear finite element approximation of the Barenblatt-Pattle solution of PME shows a first-order convergence for arclength-based adaptive meshes and a second-order convergence for Hessian-based adaptive meshes. Another feature of the method that is different from those of Duque et al. [@Duque2014; @Duque2015] and Baines et al. [@BHJ05; @BHJ05a; @Baines-2011; @BHJJ06] is that PME is solved on a large domain that contains the free boundary for the whole time period under consideration. In this way, there is no need to explicitly trace the movement of the free boundary and thus the method can readily deal with more complicated structures in the solution and in the differential equation. Numerical examples with simple free boundary (such as the Barenblatt-Pattle solution) and more complex and even merging free boundaries will be presented as well as those for PME with variable exponents and absorption.

In addition to the above mentioned MMPDE method, a number of other moving mesh methods have been developed in the past; e.g., see Hirt et al. [@HAC74] (ALE – Arbitrary Lagrangian-Eulerian) Miller and Miller [@MM81] (MFE – moving finite element), Liao and Anderson [@LA92] (deformation map), Li et al. [@LTZ01] (mesh rezoning), Cao et al. [@CHR02] (geometric conservation law), 