A Posteriori Error Estimates for the Virtual Element Method for the Stokes Problem
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A Posteriori Error Estimates for the Virtual Element Method for the Stokes Problem Gang Wang1 · Ying Wang2 · Yinnian He3 Received: 25 September 2019 / Revised: 7 June 2020 / Accepted: 25 June 2020 © Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract This paper presents a residual-type a posteriori error estimator for the virtual element method for the Stokes problem. It is proved that the a posteriori error estimator is reliable and efficient. The virtual element method allows the use of very general polygonal meshes and handles the hanging nodes naturally. Consequently, the local post-processing of locally adapted mesh can be avoided, which simplifies the adaptive procedure. A series of numerical examples are reported to show the effectiveness of adaptive mesh refinement driven by this estimator. Keywords Stokes problem · Virtual element method · A posteriori error estimate · Polygonal meshes
1 Introduction The Stokes model problem describes the steady viscous incompressible flow that has a high viscosity. If the geometric shape of the considered domain is non-convex or there is a discontinuity for the boundary data, the solution of the Stokes problem always has singularity at the concave corner or at the discontinuous point. The Stokes problem is actually a limit case of the Navier–Stokes problem, where the convective term is neglected due to the high viscosity and the flow is very slow. In terms of the Navier–Stokes problem, the vortex arises when the viscosity decreases. In these cases, there will be large errors in the numerical
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Yinnian He [email protected] Gang Wang [email protected] Ying Wang [email protected]
1
School of Mathematics and Statistics, Northwestern Polytechnical University, Xi’an 710129, Shaanxi, People’s Republic of China
2
School of Science, Xi’an University of Architecture and Technology, Xi’an 710055, People’s Republic of China
3
School of Mathematics and Statistics, Xi’an Jiaotong University, Xi’an 710049, Shaanxi, People’s Republic of China 0123456789().: V,-vol
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Journal of Scientific Computing
(2020) 84:37
solution of the finite element scheme at the places of greater concern like the singular point and the center of vortex. Since the pioneer work in [2], adaptive mesh refinement driven by a posteriori error estimators has proved to be an efficient tool to solve the scientific and engineering problems. The key of this technique is to design an efficient and reliable a posteriori error estimator using the numerical solution and the known data like the load term. At the first step, one computes the numerical solution from the discrete scheme on a coarse mesh. Secondly, one calculates the a posteriori error estimator on each element of coarse mesh. Thirdly, according to the estimators, one marks the elements having larger errors by a specific marking method. The last step is to refine the marked elements. In order to preserve the mesh conformity, further mesh refinement in the last step is required to remove han
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