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Adaptive VEM for variable data: convergence and optimality
IMA Journal of Numerical Analysis ( IF 2.3 ) Pub Date : 2023-11-16 , DOI: 10.1093/imanum/drad085
L Beirão da Veiga 1 , C Canuto 2 , R H Nochetto 3 , G Vacca 4 , M Verani 5
Affiliation  

We design an adaptive virtual element method (AVEM) of lowest order over triangular meshes with hanging nodes in 2d, which are treated as polygons. AVEM hinges on the stabilization-free a posteriori error estimators recently derived in Beirão da Veiga et al. (2023, Adaptive VEM: stabilization-free a posteriori error analysis and contraction property. SIAM J. Numer. Anal., 61, 457–494). The crucial property, which also plays a central role in this paper, is that the stabilization term can be made arbitrarily small relative to the a posteriori error estimators upon increasing the stabilization parameter. Our AVEM concatenates two modules, GALERKIN and DATA. The former deals with piecewise constant data and is shown in the above article to be a contraction between consecutive iterates. The latter approximates general data by piecewise constants to a desired accuracy. AVEM is shown to be convergent and quasi-optimal, in terms of error decay versus degrees of freedom, for solutions and data belonging to appropriate approximation classes. Numerical experiments illustrate the interplay between these two modules and provide computational evidence of optimality.

中文翻译:

适用于可变数据的自适应 VEM:收敛性和最优性

我们设计了一种在具有二维悬挂节点的三角形网格上最低阶的自适应虚拟元素方法(AVEM),将其视为多边形。AVEM 取决于 Beirão da Veiga 等人最近导出的无稳定性后验误差估计器。(2023,自适应 VEM:无稳定性后验误差分析和收缩特性。SIAM J. Numer. Anal., 61, 457–494)。在本文中也起着核心作用的关键属性是,在增加稳定参数时,稳定项可以相对于后验误差估计器任意小。我们的 AVEM 连接两个模块:GALERKIN 和 DATA。前者处理分段常数数据,并在上面的文章中显示为连续迭代之间的收缩。后者通过分段常数将一般数据近似到所需的精度。对于属于适当近似类的解和数据,AVEM 在误差衰减与自由度方面表现出收敛性和准最优性。数值实验说明了这两个模块之间的相互作用,并提供了最优性的计算证据。
更新日期:2023-11-16
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