In this work, the convergence behavior of a time-simultaneous two-grid algorithm for the one-dimensional heat equation is studied using Fourier arguments in space. The underlying linear system of equations is obtained by a finite element or finite difference approximation in space while the semi-discrete problem is discretized in time using the θ-scheme. The simultaneous treatment of all time instances leads to a global system of linear equations which provides the potential for a higher degree of parallelization of multigrid solvers due to the increased number of degrees of freedom per spatial unknown. It is shown that the all-at-once system based on an equidistant discretization in space and time stays well conditioned even if the number of blocked time-steps grows arbitrarily. Furthermore, mesh-independent convergence rates of the considered two-grid algorithm are proved by adopting classical Fourier arguments in space without assuming periodic boundary conditions. The rate of convergence with respect to the Euclidean norm does not deteriorate arbitrarily if the number of blocked time steps increases and, hence, underlines the potential of the solution algorithm under investigation. Numerical studies demonstrate why minimizing the spectral norm of the iteration matrix may be practically more relevant than improving the asymptotic rate of convergence.

中文翻译：

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对一维热方程使用阻尼 Jacobi 波形松弛平滑器的时间同时两网格算法的傅里叶分析

在这项工作中，使用傅里叶参数在空间中研究了一维热方程的时间同步两网格算法的收敛行为。基本线性方程组是通过空间中的有限元或有限差分逼近获得的，而半离散问题使用时间离散化θ-方案。所有时间实例的同时处理导致了一个全局线性方程组，由于每个空间未知数的自由度数量增加，这为多重网格求解器的更高程度的并行化提供了潜力。结果表明，即使阻塞的时间步数任意增长，基于空间和时间等距离散化的 all-at-once 系统仍保持良好状态。此外，在不假设周期性边界条件的情况下，通过在空间中采用经典傅立叶参数证明了所考虑的两网格算法的网格无关收敛速度。如果阻塞的时间步数增加，则相对于欧几里德范数的收敛速度不会任意恶化，因此强调了正在研究的解决方案算法的潜力。