Higher-order adaptive virtual element methods with contraction properties

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. Beirão 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 $k\ge 2$ 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, (squared) residual estimator, and (squared) virtual inconsistency estimator.