A posteriori error estimates for mixed virtual element methods

TL;DR

This paper develops an residual-type a posteriori error estimator for mixed virtual element methods applied to elliptic PDEs, enabling effective adaptive mesh refinement on general polygonal meshes.

Contribution

It introduces a reliable and efficient residual-based error estimator for mixed VEM, applicable to general polygonal meshes, with proven bounds and numerical validation.

Findings

Estimator is reliable and efficient for mixed VEM.

Effective in driving adaptive mesh refinement.

Numerical results confirm theoretical properties.

Abstract

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