  A structure-preserving Finite Element Method (FEM) for the transport equation
in one- and two-dimensional domains is presented. This Distributed Parameter
System (DPS) has non-collocated boundary control and observation, and reveals a
scattering-energy preserving structure. We show that the discretized model
preserves the aforementioned structure from the original infinite-dimensional
system. Moreover, we analyse the case of moving meshes for the one-dimensional
case. The moving mesh requires less states than the fixed one to produce
solutions with a comparable accuracy, and it can also reduce the overshoot and
oscillations of Gibbs phenomenon produced when using the FEM. Numerical
simulations are provided for the case of a one-dimensional transport equation
with fixed and moving meshes.
