A posteriori error estimates for the virtual element method

Abstract

An posteriori error analysis for the virtual element method (VEM) applied to general elliptic problems is presented. The resulting error estimator is of residual-type and applies on very general polygonal/polyhedral meshes. The estimator is fully computable as it relies only on quantities available from the VEM solution, namely its degrees of freedom and element-wise polynomial projection. Upper and lower bounds of the error estimator with respect to the VEM approximation error are proven. The error estimator is used to drive adaptive mesh refinement in a number of test problems. Mesh adaptation is particularly simple to implement since elements with consecutive co-planar edges/faces are allowed and, therefore, locally adapted meshes do not require any local mesh post-processing.

Gradient Recovery for the BEM‐based FEM and VEM

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A posteriori error estimate for discontinuous Galerkin finite element method on polytopal mesh

“…Now we consider the lower bound (efficiency) estimate by using standard bubble function technique. Bubble functions on general polygons can be constructed in the manner of (, page 7).Lemma (Element bubble functions ) . Let and let be the corresponding bubble function.…”

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“…The ideas we present here can, however, be generalised to much more complicated situations by applying similar processes to compute the various required terms. The possible extensions of this code are endless: the implementation of higher order methods, more general elliptic operators including lower order terms and non-constant coefficients [8,14], basis functions with higher global regularity properties [10], mesh adaptation driven by a posteriori error indicators [12], or the consideration of time dependent problems [30] to name but a few.…”