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，为求解时间上光滑周期变化的输运问题（如心肺血流问题）提供了显著优势。本文针对对流扩散方程，研究了由此产生的时间-谱格式的有限元解。除基准的伽辽金方法外，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。此外，我们引入了一种新的增广SUPG（ASU）方法，该方法通过设计在一维分段线性插值函数下能产生节点精确解。通过使用一维、二维和三维典型算例对这五种方法进行比较表明，虽然ASU整体精度最高，但在高沃默斯利数的三维极端振荡流中表现出稳定性问题。而GLS方法（在该问题中与VMS方法等价）因其优异的稳定性和合理的精度，成为一种极具吸引力的替代方案。