论文标题
抛物线PDE的不连续的Galerkin时间步进的错误概况
Error Profile for Discontinuous Galerkin Time Stepping of Parabolic PDEs
论文作者
论文摘要
我们考虑使用不连续的Galerkin（DG）方法使用分段$ \ t $中的分段多项式来考虑线性抛物线问题的时间离散化，对于$ r \ ge1 $，最大步骤尺寸〜$ k $。众所周知，DG错误的空间$ L_2 $ - 及时是最佳订单$ K^r $，并且对于$ r \ ge2 $，在节点上的$ r \ ge2 $，$ k^{2r-1} $。我们表明，在$ n $ th subinterval $（t_ {n-1}，t_n）$上，DG错误中的主要术语与本地权利radau radau radau radau $ r $。此错误配置文件意味着DG错误是每个间隔中右手高斯 - radau正交点处的$ k^{r+1} $的顺序。我们表明，左端点$ t_ {n-1} $在DG解决方案中跳跃的标准提供了准确的\ emph {a postteriori}估计值，对于子间隙$（t_ {n-1}，t_n）$的最大误差。此外，一个简单的后处理步骤产生了a \ emph {连续} $ r $的分段多项式，并具有最佳的全局收敛速率$ k^{r+1} $。我们通过一些数值实验来说明这些结果。
We consider the time discretization of a linear parabolic problem by the discontinuous Galerkin (DG) method using piecewise polynomials of degree at most $r-1$ in $t$, for $r\ge1$ and with maximum step size~$k$. It is well known that the spatial $L_2$-norm of the DG error is of optimal order $k^r$ globally in time, and is, for $r\ge2$, superconvergent of order $k^{2r-1}$ at the nodes. We show that on the $n$th subinterval $(t_{n-1},t_n)$, the dominant term in the DG error is proportional to the local right Radau polynomial of degree $r$. This error profile implies that the DG error is of order $k^{r+1}$ at the right-hand Gauss--Radau quadrature points in each interval. We show that the norm of the jump in the DG solution at the left end point $t_{n-1}$ provides an accurate \emph{a posteriori} estimate for the maximum error over the subinterval $(t_{n-1},t_n)$. Furthermore, a simple post-processing step yields a \emph{continuous} piecewise polynomial of degree $r$ with the optimal global convergence rate of order $k^{r+1}$. We illustrate these results with some numerical experiments.
