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### Moving meshes around a moving cylinder with or without concentration:

### Burgers' equation:

2D: A moving mesh of 3200 triangle elements with the two-level mesh movement strategy was used. The PDE
was solved on a finer mesh of 12800 elements.

### The radiation diffusion equation:

2D: A mesh of 6400 triangle elements was used. See, for instance, X. Yang et al.,
A moving mesh finite difference method for equilibrium radiation diffusion equations,
J. Comput. Phys. 298 (2015), 661-677 for the description of the (equilibrium) radiation-diffusion equation.

### The Regularized Long Wave (RLW) equations:

See Lu et al., An adaptive moving mesh finite element solution of the Regularized Long Wave equation, Journal of Scientific Computing 74 (2018), 122-144. (DOI: 10.1007/s10915-017-0427-6) or arXiv:1606.06541.

2D: RLW with the undularbore initial solution. A moving mesh of 144000 elements was used
on the domain (-60,300)X(-60,300).

### The Porous Medium Equation (PME):

See C. Ngo and W. Huang, A study on moving mesh finite element solution of the porous medium equation, Journal of Computational Physics 331 (2017), 357-380. (DOI: 10.1016/j.jcp.2016.11.045) or arXiv:1605.03570.

### Phase-field modeling of brittle fracture:

See F. Zhang, W. Huang, X. Li, and S. Zhang, Moving Mesh Finite Element Simulation for Phase-Field Modeling of Brittle Fracture and Convergence of Newton's Iteration, Journal of Computational Physics 356 (2018), 127-149. (DOI: 10.1016/j.jcp.2017.11.033) or (arXiv:1706.05449).