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We present an hp-adaptive virtual element method (VEM) based on thehypercircle method of Prager and Synge for the approximation of solutions todiffusion problems. We introduce a reliable and efficient a posteriori errorestimator, which is computed by solving an auxiliary global mixed problem. Weshow that the mixed VEM satisfies a discrete inf-sup condition, with inf-supconstant independent of the discretization parameters. Furthermore, weconstruct a stabilization for the mixed VEM, with explicit bounds in terms ofthe local degree of accuracy of the method. The theoretical results aresupported by several numerical experiments, including a comparison with theresidual a posteriori error estimator. The numerics exhibit the p-robustness ofthe proposed error estimator. In addition, we provide a first step towards thelocalized flux reconstruction in the virtual element framework, which leads toan additional reliable a posteriori error estimator that is computed by solvinglocal (cheap-to-solve and parallelizable) mixed problems. We providetheoretical and numerical evidence that the proposed local error estimatorsuffers from a lack of efficiency.

Adaptive virtual element methods with equilibrated fluxes