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We present an a posteriori error analysis for the mixed virtual elementmethod (mixed VEM) applied to second order elliptic equations in divergenceform with mixed boundary conditions. The resulting error estimator is ofresidual-type. It only depends on quantities directly available from the VEMsolution and applies on very general polygonal meshes. The proof of the upperbound relies on a global inf-sup condition, a suitable Helmholtz decomposition,and the local approximation properties of a Cl\'ement-type interpolant. Inturn, standard inverse inequalities and localization techniques based on bubblefunctions are the main tools yielding the lower bound. Via the inclusion of afully local postprocessing of the mixed VEM solution, we also show that theestimator provides a reliable and efficient control on the broken$\mathrm{H}(\mathrm{div})$-norm error between the exact and the postprocessedflux. Numerical examples confirm the theoretical properties of our estimator,and show that it can be effectively used to drive an adaptive mesh refinementalgorithm.

A posteriori error estimates for mixed virtual element methods