Residual a posteriori error estimation for the Virtual Element Method for elliptic problems

¹ Dipartimento di Matematica “F. Enriques”, Università degli Studi di Milano, via Saldini 50, 20133 Milano, Italy
lourenco.beirao@unimi.it
² Applied Mathematics and Plasma Physics Group, Theoretical Division, Los Alamos National Laboratory, Los Alamos, NM 87545, USA
gmanzini@lanl.gov
³ IMATI del CNR, via Ferrata 1, 27100 Pavia, Italy

Received: 3 November 2013
Revised: 30 June 2014

Abstract

A posteriori error estimation and adaptivity are very useful in the context of the virtual element and mimetic discretization methods due to the flexibility of the meshes to which these numerical schemes can be applied. Nevertheless, developing error estimators for virtual and mimetic methods is not a straightforward task due to the lack of knowledge of the basis functions. In the new virtual element setting, we develop a residual based a posteriori error estimator for the Poisson problem with (piecewise) constant coefficients, that is proven to be reliable and efficient. We moreover show the numerical performance of the proposed estimator when it is combined with an adaptive strategy for the mesh refinement.