The DPG-Star method
The DPG-Star method
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DPG-Star 方法
DOI：
10.1016/j.camwa.2020.01.012
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发表时间：
2020
期刊：
影响因子：
--
通讯作者：
Keith, B
中科院分区：
文献类型：
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作者：
Demkowicz, L;Gopalakrishnan, J;Keith, B
本文介绍DPG-star（以下简称DPG*）有限元法。该方法在某种意义上与不连续 Petrov-Galerkin (DPG) 方法是双重的。 DPG 方法可以被视为解决边值问题的超定离散化的一种方法。同样，DPG* 方法是解决欠定离散化的一种方法。这两种观点是通过将相同的算子方程嵌入到两个不同的鞍点问题中而发展起来的。对两个问题的分析有很多共同点。与文献中其他方法的比较完善了新获得的观点。值得注意的是，DPG* 和 DPG 方法可以分别视为 L L* 和最小二乘法的推广。详细考虑了 DPG* 方法的先验误差分析和后验误差控制。提供了几个数值实验的报告，证明了新方法的基本特征。 DPG* 和 DPG 分析结果之间的显着差异在于，前者的收敛速度受到无关拉格朗日乘子变量的规律性的限制。
This article introduces the DPG-star (from now on, denoted DPG*) finite element method. It is a method that is in some sense dual to the discontinuous Petrov–Galerkin (DPG) method. The DPG methodology can be viewed as a means to solve an overdetermined discretization of a boundary value problem. In the same vein, the DPG* methodology is a means to solve an underdetermined discretization. These two viewpoints are developed by embedding the same operator equation into two different saddle-point problems. The analyses of the two problems have many common elements. Comparison to other methods in the literature round out the newly garnered perspective. Notably, DPG* and DPG methods can be seen as generalizations of L L∗ and least-squares methods, respectively. A priori error analysis and a posteriori error control for the DPG* method are considered in detail. Reports of several numerical experiments are provided which demonstrate the essential features of the new method. A notable difference between the results from the DPG* and DPG analyses is that the convergence rates of the former are limited by the regularity of an extraneous Lagrange multiplier variable.