We present a priori error estimates for a multirate time-stepping scheme for coupled differential equations. The discretization is based on Galerkin methods in time using two different time meshes for two parts of the problem. We aim at surface coupled multiphysics problems like two-phase flows. Special focus is on the handling of the interface coupling to guarantee a coercive formulation as key to optimal order error estimates. In a sequence of increasing complexity, we begin with the coupling of two ordinary differential equations, coupled heat conduction equation, and finally a coupled Stokes problem. For this we show optimal multi-rate estimates in velocity and a suboptimal result in pressure. The a priori estimates prove that the multirate method decouples the two subproblems exactly. This is the basis for adaptive methods which can choose optimal lattices for the respective subproblems.

翻译：我们针对耦合微分方程组的双速率时间步进方案给出了先验误差估计。该离散化基于时间上的Galerkin方法，对问题的两部分使用了两种不同的时间网格。我们重点关注诸如两相流等表面耦合多物理场问题。特别关注界面耦合的处理方式，以确保公式的强制性，这是实现最优阶误差估计的关键。在复杂度递增的系列中，我们从两个常微分方程的耦合开始，继而研究耦合热传导方程，最终解决耦合Stokes问题。对此，我们展示了速度场的最优多速率估计以及压力场的次优结果。先验估计证明，该多速率方法能够精确解耦两个子问题。这为自适应方法奠定了基础，使得各子问题可选择最优网格。