A Space-Time Adaptive Finite Element Algorithm Based on Dual Weighted Residual Methodology for Parabolic Equations
A Space-Time Adaptive Finite Element Algorithm Based on Dual Weighted Residual Methodology for Parabolic Equations
复制标题
基于双加权残差法的抛物型方程时空自适应有限元算法
DOI：
10.1137/070698853
复制
发表时间：
2009
期刊：
影响因子：
--
通讯作者：
J. Carpio
中科院分区：
文献类型：
--
作者：
R. Bermejo;J. Carpio
本文提出了一种基于对偶加权残差（DWR）思想的时空自适应有限元算法。我们的算法包括在每个时间区间$I_n:=(t_{n-1},t_n]$局部应用DWR技术，因此，我们控制了解$J(u)$的函数的局部或截断误差。与传统的DWR方法相比，我们的方法的缺点是具有良好的局部误差控制并不能保证全局误差有界。然而，我们的方法的主要优点是我们避免了在整个时间间隔$I:=(0,T]$中向后求解相关的线性对偶问题，因此存储需求更小。这意味着我们可以定义一个自给自足的标准，允许我们控制时间步长$\Delta t$和网格尺寸h随着积分时间的推移。该算法的另一个优点是将传统DWR方法的空间后处理过程扩展到非结构化图像。
We propose in this paper a space-time adaptive algorithm based on the dual weighted residual (DWR) idea in the framework of a finite element method. Our algorithm consists of applying the DWR technique locally in each time interval $I_n:=(t_{n-1},t_n]$, and thus, we control the local or truncation error for a functional of the solution $J(u)$. The disadvantage of our method compared to the traditional DWR method is that having a good local error control does not guarantee that the global error will be bounded. However, the main advantage of our method is that we avoid solving backward in the whole time interval $I:=(0,T]$ the associated linear dual problem so that the storage requirements are smaller. This means that we can define a self-sufficient criterion that allows us to have control of the time step $\Delta t$ and the mesh size h as time of integration progresses. Another good feature of our algorithm is the extension of the spatial postprocessing procedure of the traditional DWR method to unstructu...