Analysis of the Fictitious Domain Method with L 2 -penalty for Elliptic and Parabolic Problems *
Analysis of the Fictitious Domain Method with L 2 -penalty for Elliptic and Parabolic Problems *
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椭圆和抛物线问题的 L 2 罚虚拟域方法分析 *
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发表时间：
2012
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通讯作者：
Guanyu Zhou
中科院分区：
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作者：
Guanyu Zhou
椭圆和抛物问题的L2罚虚拟区域法分析.分别考虑椭圆型和抛物型问题的带L2罚的虚拟区域方法.给出了L2-罚问题的正则性定理和先验估计.给出了罚元和P1元有限元插值的误差估计.数值实验证实了理论结果。1.导论.本文的目的是建立椭圆和抛物问题的虚拟区域方法的数学研究。众所周知，虚拟域方法是基于在一个更大的空间域（称为虚拟域）中对原始问题进行重新表述，该空间域具有简单的形状。然后，虚拟域可以离散成一个均匀的网格，与原始边界无关。这种方法的优点是，我们可以避免一个边界拟合网格的耗时的建设。此外，这种方法将有助于解决时间相关的移动边界问题。在我们以前的报告（[14，15]）中，我们发展了一个求解椭圆和抛物问题的H1-罚虚拟区域方法的数学理论。本文的目的是建立严格的估计所引起的L2惩罚和有限元插值的错误。我们通过研究H2正则性和L2罚问题的估计来研究L2罚，这是与[1]不同的方法，其中在没有数值分析的情况下考虑Navier-Stokes方程的L2罚。由于我们的正则性和估计结果，使得有限元分析变得容易处理，我们对椭圆和抛物问题的L2罚的H1模的误差估计保持了文[1]中对Navier-Stokes问题的误差估计的尖锐性，并且给出了L2模的误差估计.文献[7]证明了椭圆和抛物问题的L2罚函数的收敛性，但没有给出误差估计，有限元分析也没有给出。我们的分析方法也可以应用到Stokes和Navier-Stokes问题，几乎没有困难。本文的其余部分安排如下。在第2节中，我们考虑椭圆问题。我们首先给出了L2惩罚的误差估计，然后我们转向有限元逼近。第三节是专门讨论抛物问题，同样的方式对椭圆的情况。数值实验来验证...
Analysis of the fictitious domain method with L 2-penalty for elliptic and parabolic problems Abstract. The fictitious domain method with L 2-penalty for elliptic and parabolic problems are considered, respectively. The regularity theorems and a priori estimates for L 2-penalty problems are given. We derive error estimates for penalization and finite element interpolation with P 1-element. Numerical experiments are performed to confirm the theoretical results. 1. Introduction. The purpose of this paper is to establish a mathematical study of the fictitious domain method for elliptic and parabolic problems. The fictitious domain method is well known to be based on a reformulation of the original problem in a larger spatial domain, called the fictitious domain, with a simple shape. Then, the fictitious domain can be discretized by a uniform mesh, independent of the original boundary. The advantage of this approach is that we can avoid the time-consuming construction of a boundary-fitted mesh. Furthermore, this approach will be useful to solve time-dependent moving-boundary problems. In our previous reports ([14, 15]), we developed a mathematical theory for the H 1-penalty fictitious domain method for elliptic and parabolic problems. The aim of this paper is to establish rigorous estimates of the errors induced by L 2 penalization and finite element interpolation. We examine the L 2 penalization by studying the H 2 regularity and estimates of the L 2-penalty problem, which is a different approach from [1], where the L 2 penalization for Navier-Stokes equation is considered without numerical analysis. Thanks to our regularity and estimate results, the finite element analysis becomes easy to treat. Our error estimates in the H 1 norm of L 2 penalization for elliptic and parabolic problems maintain the sharpness of those for Navier-Stokes problems in [1]; moreover, we show the error estimates of L 2 norm. The convergence of L 2 penalization for elliptic and parabolic problems has been proved in [7]; however, no error estimate has been found, neither the finite element analysis. Our analysis method presented here can also be applied to Stokes and Navier-Stokes problems with little difficulty. The rest of this paper is arranged as follow. In Sect. 2, we consider the elliptic problem. We first show the error estimates for L 2 penalization, then we turn to the finite element approximation. And Sect. 3 is devoted to the parabolic problem, as the same way to the elliptic case. The numerical experiments to validate …