Stability Analysis for the Virtual Element Method
Stability Analysis for the Virtual Element Method
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DOI：
10.1142/s021820251750052x
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发表时间：
2016-07
期刊：
影响因子：
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通讯作者：
L. Veiga;C. Lovadina;A. Russo
中科院分区：
文献类型：
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作者：
L. Veiga;C. Lovadina;A. Russo
我们分析了一个简单的椭圆模型问题的虚元法（VEM），允许更一般的网格比一个通常认为在VEM文献。例如，可以处理具有任意小边（相对于父元素直径）的网格。我们的一般方法适用于不同的选择的稳定性形式，包括，例如，“经典”的一个介绍了[L。Beirao da Veiga，F. Brezzi，A. Cangiani，G.曼齐尼湖D.马里尼和A.徐文，虚单元法的基本原理，数学模型与方法，应用科学。23（2013），no. 1，199-214]，以及最近的一个在[Wriggers，P.，拉斯特，W.T.，和Reddy，B. D.，一种用于接触的虚元素方法，已提交出版]。最后，我们表明，稳定项可以简化下降的内部元素的自由度的贡献。由此产生的稳定形式，只涉及边界自由度，可以使用在VEM计划，而不影响稳定性和收敛性能。数值试验结果与理论预测一致。
We analyse the Virtual Element Methods (VEM) on a simple elliptic model problem, allowing for more general meshes than the one typically considered in the VEM literature. For instance, meshes with arbitrarily small edges (with respect to the parent element diameter), can be dealt with. Our general approach applies to different choices of the stability form, including, for example, the "classical" one introduced in [L. Beirao da Veiga, F. Brezzi, A. Cangiani, G. Manzini, L. D. Marini, and A. Russo, Basic principles of virtual element methods, Math. Models Methods Appl. Sci. 23 (2013), no. 1, 199-214], and a recent one presented in [Wriggers, P., Rust, W.T., and Reddy, B.D., A virtual element method for contact, submitted for publication]. Finally, we show that the stabilization term can be simplified by dropping the contribution of the internal-to-the-element degrees of freedom. The resulting stabilization form, involving only the boundary degrees of freedom, can be used in the VEM scheme without affecting the stability and convergence properties. The numerical tests are in accordance with the theoretical predictions.