On the necessity of Nitsche term
On the necessity of Nitsche term
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DOI：
10.1016/j.apnum.2010.04.013
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发表时间：
2010-09
影响因子：
2.8
通讯作者：
Gaël Dupire;J. Boufflet;M. Dambrine;Pierre Villon
中科院分区：
文献类型：
--
作者：
Gaël Dupire;J. Boufflet;M. Dambrine;Pierre Villon
本文的目的是探索用一族固定的有限元形函数族求解Dirichlet边值问题的可能性，并给出一种变分形式。该区域被嵌入到边界框中，并且有限元近似与该框的规则结构网格相关联。区域的形状与离散化网格无关。在这些条件下，永远不需要啮合工具。这在不断发展的域的情况下可能特别有用，例如形状优化或移动界面。这并不是一个新想法，但我们在这里分析了一种特殊的方法。这种方法的主要困难是相关的二次型不是强制的，并且必须检查inf-sup条件。在一维中，我们证明了该公式是适定的，并给出了误差估计。然而，我们依赖于显式计算的证明仅限于这种情况，并且我们在第二维给出了数值证据，证明该公式不提供可靠的方法。我们首先通过Nitsche项添加正则化，并且我们观察到仍然存在一些不稳定性。然后，我们引入并证明几何正则化。利用这两种正则化方法得到了一种可靠的方法。
The aim of this article is to explore the possibility of using a family of fixed finite elements shape functions to solve a Dirichlet boundary value problem with an alternative variational formulation. The domain is embedded in a bounding box and the finite element approximation is associated to a regular structured mesh of the box. The shape of the domain is independent of the discretization mesh. In these conditions, a meshing tool is never required. This may be especially useful in the case of evolving domains, for example shape optimization or moving interfaces. This is not a new idea, but we analyze here a special approach. The main difficulty of the approach is that the associated quadratic form is not coercive and an inf-sup condition has to be checked. In dimension one, we prove that this formulation is well posed and we provide error estimates. Nevertheless, our proof relying on explicit computations is limited to that case and we give numerical evidence in dimension two that the formulation does not provide a reliable method. We first add a regularization through a Nitsche term and we observe that some instabilities still remain. We then introduce and justify a geometrical regularization. A reliable method is obtained using both regularizations.