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Least‐squares virtual element method for the convection‐diffusion‐reaction problem
Author(s) -
Wang Gang,
Wang Ying,
He Yinnian
Publication year - 2021
Publication title -
international journal for numerical methods in engineering
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.421
H-Index - 168
eISSN - 1097-0207
pISSN - 0029-5981
DOI - 10.1002/nme.6636
Subject(s) - a priori and a posteriori , estimator , finite element method , polygon mesh , least squares function approximation , convection–diffusion equation , mathematical optimization , mathematics , flexibility (engineering) , algorithm , computer science , boundary (topology) , mesh generation , element (criminal law) , diffusion , geometry , mathematical analysis , engineering , statistics , structural engineering , physics , thermodynamics , philosophy , epistemology , law , political science
In this paper, we introduce a least‐squares virtual element method for the convection‐diffusion‐reaction problem in mixed form. We use the H (div) virtual element and continuous virtual element to approximate the flux and the primal variables, respectively. The method allows for the use of very general polygonal meshes. Optimal order a priori error estimates are established for the flux and the primal variables in suitable norms. The least‐squares method offers an efficient a posteriori error estimator without extra effort. Moreover, the hanging nodes are naturally treated in the virtual element method, which provides the high flexibility in mesh refinement because the local mesh postprocessing is never required. Both attractive features motivate us to develop the a posteriori error estimate of the method. Numerical experiments are shown to illustrate the accuracy of the theoretical analysis and demonstrate that the adaptive mesh refinement driven by the proposed estimator can efficiently capture the boundary and the interior layers.