A posteriori error estimates for the virtual element method

A posteriori error estimates for a Virtual Element Method for the Steklov eigenvalue problem

“…As a consequence of this replacement, new additional terms appear in the a posteriori error estimator, which represent the virtual inconsistency of VEM. Similar terms also appear in the other papers for a posteriori error estimates of VEM (see [11,21]). We prove that the error estimator is equivalent to the error and use the corresponding indicator to drive an adaptive scheme.…”

“…We introduce bubble functions on polygons as follows (see [21]). An interior bubble function ψ K ∈ H 1 0 (K) for a polygon K can be constructed piecewise as the sum of the cubic bubble functions for each triangle of the sub-triangulation T K h that attain the value 1 at the barycenter of each triangle.…”

“…The indicators η K include the terms θ K which do not appear in standard finite element estimators. This term, which represent the virtual inconsistency of the method, has been introduced in [11,21] for a posteriori error estimates of other VEM. Let us emphasize that it can be directly computed in terms of the bilinear form S K (•, •).…”

“…On the other hand, the design and analysis of a posteriori error bounds for the VEM is a challenging task. References [11,21] are the only a posteriori error analyses for VEM currently available in the literature. In [11], a posteriori error bounds for the C 1 -conforming VEM for the two-dimensional Poisson problem are proposed.…”

“…In [11], a posteriori error bounds for the C 1 -conforming VEM for the two-dimensional Poisson problem are proposed. In [21], a posteriori error bounds are introduced for the C 0 -conforming VEM proposed in [22] for the discretization of second order linear elliptic reaction-convection-diffusion problems with non constant coefficients in two and three dimensions.…”