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.

翻译：本文提出了一种适用于一维和二维域中输运方程的保结构有限元方法。该分布参数系统具有非同位边界控制与观测特性，并展现出散射-能量守恒结构。我们证明离散化模型保留了原始无限维系统的上述结构。此外，针对一维情形分析了随动网格的情况。与固定网格相比，随动网格在生成同等精度解时所需状态更少，并且能有效降低有限元法产生的吉布斯现象的过冲与振荡。针对一维输运方程，给出了固定网格与随动网格的数值仿真结果。