Anisotropic a posteriori error estimate for the virtual element method

Abstract

We derive an anisotropic a posteriori error estimate for the adaptive conforming virtual element approximation of a paradigmatic two-dimensional elliptic problem. In particular, we introduce a quasi-interpolant operator and exploit its approximation results to prove the reliability of the error indicator. We design and implement the corresponding adaptive polygonal anisotropic algorithm. Several numerical tests assess the superiority of the proposed algorithm in comparison with standard polygonal isotropic mesh refinement schemes.

On arbitrarily regular conforming virtual element methods for elliptic partial differential equations

“…In this section, we briefly recall a convergence result in the energy norm [10] (see also [4,46]) for the approximation of (1a)-(1b). In particular, employing Theorem 3.1 together with standard results of approximation (see, e.g., Reference [19,46,60]) and the approximation properties of the right-hand side contained in Section 4.7.…”

“…Remark 4.18. Convergence estimates in lower order norms can be established provided that classical duality arguments can be used and that the polynomial approximation order 𝑟 is sufficient large [4,10,38].…”

Comparison of standard and stabilization free Virtual Elements on anisotropic elliptic problems

“…The problem is discretized with bilinear forms that consist of a polynomial part that mimics the operator and an arbitrary stabilizing bilinear form. In [2], error analysis focused on anisotropic elliptic problems shows that the stabilization term adds an isotropic component of the error, independently of the nature of the problem. In [5], a modified version of the method, E 2 VEM, was proposed, designed to allow the definition of coercive bilinear forms that consist only of a polynomial approximation of the problem operator.…”

Lowest order stabilization free Virtual Element Method for the Poisson equation

“…The first one that we mention is the derivation of a posteriori error estimates [25,15], where the stabilization term is always at the right-hand side when bounding the error in terms of the error estimator, both from above and from below. Moreover, the isotropic nature of the stabilization term becomes an issue when devising SUPG stabilizations [13,16], or in the derivation of anisotropic a posteriori error estimates [3]. Finally, other contexts in which the stabilization may induce problems are multigrid analysis [4] and complex non-linear problems [34].…”