control the size of mesh elements but also their shape and orientation. Generally speaking, a so generated mesh has better alignment with the geometry of the physical solution than that with a scalar mesh adaptation function. Moreover, a newly developed compact formulation of the method (cf. (\[mmpde-2\]) and (\[mmpde-3\])) makes its implementation much easier and more efficient. Mesh adaptation based on the gradient and Hessian of the solution will be considered. The arclength metric tensor (a gradient-based metric tensor) has been widely used in the context of moving mesh methods because it results in more stable mesh movement than a Hessian-based metric tensor and works well for many problems. On the other hand, there is no theoretical guarantee that the arclength metric tensor will lead to the optimal convergence order for piecewise linear interpolation or finite element approximation since the error is determined by the Hessian of the solution in these situations. As a matter of fact, there are problems, although not very common, for which we have to use a Hessian-based metric tensor in order to achieve the optimal convergence order for piecewise linear interpolation or finite element approximation. Interestingly, PME is one of those problems. We shall show 