Discretizing a solution in the Fourier domain rather than the time domain presents a significant advantage in solving transport problems that vary smoothly and periodically in time, such as cardiorespiratory flows. The finite element solution of the resulting time-spectral formulation is investigated here for the convection-diffusion equations. In addition to the baseline Galerkin's method, we consider stabilized approaches inspired by the streamline upwind Petrov/Galerkin (SUPG), Galerkin/least square (GLS), and variational multiscale (VMS) methods. We also introduce a new augmented SUPG (ASU) method that, by design, produces a nodally exact solution in one dimension for piecewise linear interpolation functions. Comparing these five methods using 1D, 2D, and 3D canonical test cases shows while the ASU is most accurate overall, it exhibits stability issues in extremely oscillatory flows with a high Womersley number in 3D. The GLS method, which is identical to the VMS for this problem, presents an attractive alternative due to its excellent stability and reasonable accuracy.

翻译：在傅里叶域而非时间域中对解进行离散化，对于求解随时间平滑周期变化的输运问题（如心肺血流）具有显著优势。本文针对对流扩散方程，研究了由此产生的时间谱公式的有限元解。除经典伽辽金方法外，我们考虑了基于流线迎风彼得罗夫-伽辽金（SUPG）、伽辽金最小二乘（GLS）和变分多尺度（VMS）方法的稳定化途径。同时，我们提出一种新的增强型SUPG（ASU）方法，该方法通过设计，在使用分段线性插值函数的一维问题中可实现节点精确解。通过一维、二维和三维标准算例对五种方法进行比较表明：ASU方法整体精度最高，但在三维高沃默斯利数极端振荡流中表现出稳定性问题；与此问题中与VMS方法相同的GLS方法，因其极佳的稳定性和合理的精度，成为具有吸引力的替代方案。