DISCONTINUOUS GALERKIN FINITE ELEMENT METHOD WITH INTERIOR PENALTIES FOR CONVECTION DIFFUSION OPTIMAL CONTROL PROBLEM
DISCONTINUOUS GALERKIN FINITE ELEMENT METHOD WITH INTERIOR PENALTIES FOR CONVECTION DIFFUSION OPTIMAL CONTROL PROBLEM
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发表时间：
2009
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通讯作者：
T. Sun;Wenbin Liu;T. Sun
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作者：
T. Sun;Wenbin Liu;T. Sun
最优控制问题的有限元逼近一直是工程设计工作中的一个重要课题。对于各种最优控制问题的标准有限元逼近已经进行了广泛的理论和数值研究。例如，对于由某些线性椭圆或抛物状态方程控制的最优控制问题，很早以前就建立了有限元近似的先验误差估计，参见[1,2,3,4,5]。此外，对[6]中一些重要的流动控制问题的有限元逼近建立了先验误差估计。先验误差估计的一些最新进展可以在[7,8]和[9,10,11,12]中找到，用于后验误差估计。系统地介绍了偏微分方程的有限元方法和最优控制问题，例如[13]，[14]和[15]。近年来，不连续Galerkin方法已被证明在解决大范围的计算流体问题方面非常有用（[16,17,18]）。它们优于标准的连续伽辽金方法，因为它们在近似全局粗糙解方面具有灵活性，它们的局部质量守恒，它们在非结构化网格上的可能定义，它们的误差控制和网格自适应的潜力。在有限元法中使用惩罚项的想法并不新鲜。Baker[19]是第一个在椭圆方程中使用非协调元内罚的人。Douglas和Dupont[20]分析了一种对线性椭圆型和抛物型问题的导数进行内惩罚的方法。受[19]的启发，Wheeler[21]提出了二阶线性椭圆方程的内罚方法。与[21]最接近的是，Arnold[22]给出了二阶非线性抛物方程的半离散不连续Galerkin方法。
Finite element approximation of optimal control problems has been an important topic in engineering design work. There has been extensive theoretical and numerical studies for standard finite element approximation of various optimal control problems. For instance, for the optimal control problems governed by some linear elliptic or parabolic state equations, a priori error estimates of the finite element approximation were established long ago, see [1, 2, 3, 4, 5]. Furthermore, a priori error estimates were established for the finite element approximation of some important flow control problems in [6]. Some recent progress in a priori error estimates can be found in [7, 8] and in [9, 10, 11, 12], for a posteriori error estimates. Systematic introduction of the finite element method for PDEs and optimal control problems can be found in, for example, [13], [14] and [15]. In recent years, the discontinuous Galerkin methods have been proved very useful in solving a large range of computational fluid problems ([16, 17, 18]). They are preferred over standard continuous Galerkin methods because of their flexibility in approximating globally rough solutions, their local mass conservation, their possible definition on unstructured meshes, their potential for error control and mesh adaptation. The idea of using penalty terms in a finite element method is not new. Baker [19] was the first one who used interior penalty with nonconforming elements for elliptic equations. Douglas and Dupont [20] analyzed a method which used interior penalties on the derivatives with conforming elements for linear elliptic and parabolic problems. Inspired by [19], Wheeler [21] presented an interior penalty method for second order linear elliptic equations. Closest to [21], Arnold [22] formulated a semi-discrete discontinuous Galerkin method with interior penalty for second order nonlinear parabolic equations.