A posteriori error estimation and adaptivity in hp virtual elements

An explicit and computable error estimator for the $$hp$$ version of the virtual element method (VEM), together with lower and upper bounds with respect to the exact energy error, is presented. Such error estimator is employed to provide, following the approach of Melenk and Wohlmuth (Adv Comput Math 15(1–4):311–331, 2001), $$hp$$ adaptive mesh refinements for very general polygonal meshes. In addition, a novel VEM $$hp$$ Clément quasi-interpolant, instrumental for the a posteriori error analysis, is introduced. The performances of the adaptive method are validated by a number of numerical experiments.