Analysis of the parareal approach based on discontinuous Galerkin method for time‐dependent Stokes equations

Abstract

This paper analyzes a parareal approach based on discontinuous Galerkin (DG) method for the time-dependent Stokes equations. A class of primal discontinuous Galerkin methods, namely variations of interior penalty methods, are adopted for the spatial discretization in the parareal algorithm (we call it parareal DG algorithm). We study three discontinuous Galerkin methods for the time-dependent Stokes equations, and the optimal continuous in time error estimates for the velocities and pressure are derived. Based on these error estimates, the proposed parareal DG algorithm is proved to be unconditionally stable and bounded by the error of discontinuous Galerkin discretization after a finite number of iterations. Finally, some numerical experiments are conducted which confirm our theoretical results, meanwhile, the efficiency of the parareal DG algorithm can be seen through a parallel experiment.

Superconvergence of local discontinuous Galerkin methods with generalized alternating fluxes for 1D linear convection-diffusion equations

“…It is an attractive property, but it is also the main difficulty since the temporary accuracy should more or less match the spatial accuracy. A first numerical result has suggested that if the time discretization is under-resolved, then the spatial superconvergence will be concealed, see [15]. In this paper, we first demonstrate that for the high-order case (p ≥ 2), to benefit from the spatial superconvergence, explicit Runge-Kutta methods will suffer from an extremely small time step.…”

“…The main focus of this paper is the SDG method, which has been introduced recently by Li, Benedusi and Krause [15]. The SDG method is an iterative method derived from the DG method for time discretization, with the order of accuracy increased by one for each additional iteration.…”

“…Respect to the post-processing technique, attractively, the SDG method can be constructed systematically for the arbitrary order of accuracy, and its explicit scheme has competitive stability compared existed time integrators. In the following, we first briefly present this method as described in [15], and then provide a derivation of its explicit scheme from the point of view of a correction method. Consider the ODE system…”

“…, is an approximation of L (see [15]). Finally, we give the algorithm of the explicit SDG method in Algorithm 1.…”

“…This paper focuses on the SDG method recently constructed by Li, Benedusi, and Krause [15]. The SDG method is an iterative method for time discretization derived from the DG method, with the order of accuracy increased by one for each additional iteration.…”