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), least square (LSQ), 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 convergence issues in extremely oscillatory flows with a high Womersley number in 3D. The VMS method presents an attractive alternative due to its excellent convergence characteristics and reasonable accuracy.

翻译：在傅里叶域而非时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（如心肺血流问题）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-谱格式的有限元解。在基准伽辽金方法基础上，我们考虑了受流线迎风彼得罗夫/伽辽金法（SUPG）、最小二乘法（LSQ）和变分多尺度法（VMS）启发的稳定化方法。我们还提出了一种新的增广SUPG（ASU）方法，该方法通过设计，在一维情况下对分段线性插值函数产生节点精确解。通过使用一维、二维和三维标准算例对这五种方法进行比较，结果表明：虽然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收敛问题。VMS方法因其优异的收敛特性和合理的精度，成为颇具吸引力的替代方案。