present in the projected initial data. Numerical results show that their method gives a second-order convergence for $m=1$ but only a first-order convergence for $m=3$ when a uniform initial mesh is used [@BHJ05a; @Baines-2011]. For $m=3$, the second-order convergence can be recovered in one dimension if an optimal initial mesh is used. Unfortunately, such an optimal mesh is significantly more expensive to compute in two dimensions than in one dimension. Recently, Duque et al. [@Duque2014; @Duque2015] present a moving mesh finite element method based on an MMPDE for PME with variable exponents and with/without absorption. The method shows a first-order convergence when tested for the Barenblatt-Pattle solution of PME.

The objective of this paper is to study an adaptive moving mesh finite element method for the numerical solution of PME. The method is also based on an MMPDE but significantly different from the method of [@Duque2014; @Duque2015]. The MMPDE we use is formulated by minimizing an energy (cf. (\[Ih\])) based on the equidistribution and alignment conditions and the mesh adaptation is controlled through a matrix-valued function (i.e., a metric tensor) instead of a scalar function. The advantage of using a metric tensor is that it provides information not only to 