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：在傅里叶域而非时域中对解进行离散化，为处理时间上平滑周期变化的输运问题（如心肺血流）提供了显著优势。本文针对对流扩散方程，研究了时间-谱格式的有限元解法。除基准的伽辽金方法外，我们还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）方法的稳定化方案。此外，我们提出了一种新的增强型SUPG（ASU）方法，该方法通过设计能在分段线性插值函数的一维问题中产生节点精确解。通过一维、二维和三维典型算例对比这五种方法表明：尽管ASU方法整体精度最高，但在三维高沃默斯利数的极端振荡流中会出现稳定性问题；而GLS方法（在该问题中与VMS等价）凭借其优异的稳定性和合理的精度，成为颇具吸引力的替代方案。