by a standard time step size selection procedure (e.g., see Hairer et al. [@HNW93 Section II.4]) and using a two-step error estimator of Gonzalez-Pinto et al. [@Montijano2004]. The relative and absolute tolerances $rtol = 10^{-6}$ and $atol = 10^{-8}$ are taken in the computation.

The whole computation alternates between the integration of PME and the generation of the mesh. Starting with the current mesh $\cTh^n$ and a solution $u_h^n({\boldsymbol{x}})\approx u({\boldsymbol{x}}, t_n)$ defined thereon, a new mesh $\cTh^{n+1}$ is generated using the moving mesh strategy to be described in the next subsection. Then, the discrete PME (\[fem-2\]) is integrated from $t_n$ to $t_{n+1}$ (as described above) to obtain the solution approximation $u_h^{n+1}({\boldsymbol{x}})$.

An MMPDE-based moving mesh strategy {#SEC:mmpde}
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We now describe the generation of $\cTh^n$, $ n = 1, ..., n_f$. We assume that the mesh $\cTh^n$ and a computed solution $u_h^n({\boldsymbol{x}})$ are known at $t = t_n$. We also assume that a reference computational mesh $\cTco = \{ \hat{{\boldsymbol{\xi}}}_j, j = 1, ..., N_v\}$ having the same connectivity and the same numbers of vertices and elements as $\cTh^n$ has been chosen. In our computation, it is taken as a uniform mesh (in the Euclidean metric) defined on $\Omega$. $\cTco$ 