The realization of a standard Adaptive Finite Element Method (AFEM) preserves the mesh conformity by performing a completion step in the refinement loop: in addition to elements marked for refinement due to their contribution to the global error estimator, other elements are refined. In the new perspective opened by the introduction of Virtual Element Methods (VEM), elements with hanging nodes can be viewed as polygons with aligned edges, carrying virtual functions together with standard polynomial functions. The potential advantage is that all activated degrees of freedom are motivated by error reduction, not just by geometric reasons. This point of view is at the basis of the paper [L. Beirao da Veiga et al., Adaptive VEM: stabilization-free a posteriori error analysis and contraction property, SIAM Journal on Numerical Analysis, vol. 61, 2023], devoted to the convergence analysis of an adaptive VEM generated by the successive newest-vertex bisections of triangular elements without applying completion, in the lowest-order case (polynomial degree k=1). The purpose of this paper is to extend these results to the case of VEMs of order k>1 built on triangular meshes. The problem at hand is a variable-coefficient, second-order self-adjoint elliptic equation with Dirichlet boundary conditions; the data of the problem are assumed to be piecewise polynomials of degree k-1. By extending the concept of global index of a hanging node, under an admissibility assumption of the mesh, we derive a stabilization-free a posteriori error estimator. This is the sum of residual-type terms and certain virtual inconsistency terms (which vanish for k=1). We define an adaptive VEM of order k based on this estimator, and we prove its convergence by establishing a contraction result for a linear combination of (squared) energy norm of the error, residual estimator, and virtual inconsistency estimator.

翻译：标准自适应有限元方法（AFEM）通过在细化循环中执行完备化步骤来保持网格相容性：除因对全局误差估计有贡献而被标记细化的单元外，还需细化其他单元。在虚拟单元方法（VEM）引入所开辟的新视角下，带有悬挂节点的单元可被视为具有对齐边的多边形，其承载虚拟函数和标准多项式函数。潜在优势在于所有激活自由度均源于误差缩减需求，而非仅几何原因。这一观点构成了论文[L. Beirao da Veiga等人，自适应VEM：无稳定化的后验误差分析与收缩性质，SIAM数值分析杂志，第61卷，2023年]的基础，该文致力于研究最低阶情况（多项式阶数k=1）下，通过逐次最新顶点二分法生成三角形单元的自适应VEM（无需完备化步骤）的收敛性分析。本文旨在将上述结果推广至基于三角形网格构建的k>1阶VEM。所研究的问题是带有狄利克雷边界条件的变系数二阶自伴椭圆方程，问题数据假设为k-1阶分片多项式。通过扩展悬挂节点全局指标的概念，在网格可容许性假设下，我们推导出无需稳定化的后验误差估计量。该估计量由残差型项和某些虚拟不一致项（当k=1时该项为零）之和构成。基于该估计量我们定义了k阶自适应VEM，并通过建立误差的（平方）能量范数、残差估计量和虚拟不一致估计量的线性组合的收缩结果，证明了该方法的收敛性。