We study the impact of spatial coarsening on the convergence of the Parareal algorithm, both theoretically and numerically. For initial value problems with a normal system matrix, we prove a lower bound for the Euclidean norm of the iteration matrix. When there is no physical or numerical diffusion, an immediate consequence is that the norm of the iteration matrix cannot be smaller than unoty as soon as the coarse problem has fewer degrees-of-freedom than the fine. This prevents a theoretical guarantee for monotonic convergence, which is necessary to obtain meaningful speedups. For diffusive problems, in the worst-case where the iteration error contracts only as fast as the powers of the iteration matrix norm, making Parareal as accurate as the fine method will take about as many iterations as there are processors, making meaningful speedup impossible. Numerical examples with a non-normal system matrix show that for diffusive problems good speedup is possible, but that for non-diffusive problems the negative impact of spatial coarsening on convergence is big.

翻译：我们从理论上和数字上研究空间粗化对半官方算法趋同的影响。 对于与正常系统矩阵的初始值问题,我们证明对迭代矩阵的Euclidean规范的界限较低。当没有物理或数字扩散时,直接的后果是,迭代矩阵的规范不能比不公开的问题比罚款少自由度时小得多。这妨碍了单调趋同的理论保证,而单调趋同对于取得有意义的加速是必要的。对于最坏的情况,即迭代矩阵规范的功率只有超快的递减错误,使Paralime与精细方法一样精确,因为有处理器,使得有意义的加速是不可能的。非正常系统矩阵的数值实例表明,对于调和问题来说,良好的加速是可能的,但是对于非重叠问题来说,空间相容变异对趋同的负面影响是巨大的。