PME with a variable exponent $m = \gamma({\boldsymbol{x}})$. Zhang and Wu [@Zhang-2009] consider the numerical simulation of the one-dimensional PME on a uniform mesh using a high-order local discontinuous Galerkin finite element method. The method can effectively eliminate unwanted, nonphysical oscillations in the computed solution near the free boundary and lead to a high-order convergence rate within the solution support and away from the free boundary.

The low regularity (steep gradient and corner shape near the moving free boundary) and evolving nature of the solution makes adaptive moving mesh methods an attractive tool to improve accuracy in the numerical solution of PME. A number of works exist in this direction. For example, Budd et al. [@BCHR99] investigate the numerical simulation of self-similarity solutions of one-dimensional PME using the Moving Mesh PDE (MMPDE) moving mesh method [@HRR94a; @HR11] and a specially designed monitor function to preserve the scaling invariance of PME. In a series of papers [@BHJ05; @BHJ05a; @BHJJ06] (also see the review paper [@Baines-2011]), Baines and his co-workers study the numerical solution of PME in one and two dimensions using a moving mesh finite element method that is based upon conserving a local proportion of the total mass that is 