We present an hp-adaptive virtual element method (VEM) based on the hypercircle method of Prager and Synge for the approximation of solutions to diffusion problems. We introduce a reliable and efficient a posteriori error estimator, which is computed by solving an auxiliary global mixed problem. We show that the mixed VEM satisfies a discrete inf-sup condition with inf-sup constant independent of the discretization parameters. Furthermore, we construct a stabilization for the mixed VEM with explicit bounds in terms of the local degree of accuracy of the method. The theoretical results are supported by several numerical experiments, including a comparison with the residual a posteriori error estimator. The numerics exhibit the p-robustness of the proposed error estimator. In addition, we provide a first step towards the localized flux reconstruction in the virtual element framework, which leads to an additional reliable a posteriori error estimator that is computed by solving local (cheap-to-solve and parallelizable) mixed problems. We provide theoretical and numerical evidence that the proposed local error estimator suffers from a lack of efficiency.

Dassi, F., Gedicke, J., Mascotto, L. (2022). Adaptive virtual element methods with equilibrated fluxes. APPLIED NUMERICAL MATHEMATICS, 173, 249-278 [10.1016/j.apnum.2021.11.015].

Adaptive virtual element methods with equilibrated fluxes

Dassi, F.;Mascotto, L.
2022

Abstract

We present an hp-adaptive virtual element method (VEM) based on the hypercircle method of Prager and Synge for the approximation of solutions to diffusion problems. We introduce a reliable and efficient a posteriori error estimator, which is computed by solving an auxiliary global mixed problem. We show that the mixed VEM satisfies a discrete inf-sup condition with inf-sup constant independent of the discretization parameters. Furthermore, we construct a stabilization for the mixed VEM with explicit bounds in terms of the local degree of accuracy of the method. The theoretical results are supported by several numerical experiments, including a comparison with the residual a posteriori error estimator. The numerics exhibit the p-robustness of the proposed error estimator. In addition, we provide a first step towards the localized flux reconstruction in the virtual element framework, which leads to an additional reliable a posteriori error estimator that is computed by solving local (cheap-to-solve and parallelizable) mixed problems. We provide theoretical and numerical evidence that the proposed local error estimator suffers from a lack of efficiency.
Articolo in rivista - Articolo scientifico
Equilibrated fluxes; hp-adaptivity; Hypercircle method; Polygonal meshes; Virtual element method;
English
8-dic-2021
2022
173
249
278
none
Dassi, F., Gedicke, J., Mascotto, L. (2022). Adaptive virtual element methods with equilibrated fluxes. APPLIED NUMERICAL MATHEMATICS, 173, 249-278 [10.1016/j.apnum.2021.11.015].
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/10281/340208
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