Adaptive virtual element methods with equilibrated flux
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We present an hp-adaptive virtual element method (VEM) based on the hypercircle method of Prager and Synge for the approximation of solutions to diffusion problems. We introduce a reliable and efficient a posteriori error estimator, which is computed by solving an auxiliary global mixed problem. We show that the mixed VEM satisfies a discrete inf-sup condition, with inf-sup constant independent of the discretization parameters. Furthermore, we construct a stabilization for the mixed VEM, with explicit bounds in terms of the local degree of accuracy of the method. The theoretical results are supported by several numerical experiments, including a comparison with the residual a posteriori error estimator. The numerics exhibit the p-robustness of the proposed error estimator. In addition, we provide a first step towards the localized flux reconstruction in the virtual element framework, which leads to an additional reliable a posteriori error estimator that is computed by solving local (cheap-to-solve and parallelizable) mixed problems.
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