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 [8]. The crucial property, that 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 [8] 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.

翻译：我们针对二维三角形网格中具有悬挂节点的多边形单元，设计了最低阶自适应虚拟单元方法（AVEM）。该方法基于文献[8]近期提出的无稳定化后验误差估计子。其关键性质（这在本文中也起核心作用）是：随着稳定化参数增大，稳定项相对后验误差估计子可被任意缩减。我们的AVEM串联了两个模块：GALERKIN与DATA。前者处理分片常数数据，文献[8]已证明其在连续迭代步之间具有压缩性；后者通过分片常数对一般数据进行指定精度逼近。对于属于适当逼近类的解与数据，AVEM在误差随自由度衰减方面被证明具有收敛性与拟最优性。数值实验展示了两个模块的相互作用，并提供了最优性的计算证据。