A posteriori error estimates of mixed discontinuous Galerkin method for a class of Stokes eigenvalue problems

Lingling Sun; Hai Bi; Yidu Yang; Lingling Sun; Hai Bi; Yidu Yang

doi:10.3934/math.20231084

AIMS Mathematics

2023, Volume 8, Issue 9: 21270-21297. doi: 10.3934/math.20231084

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A posteriori error estimates of mixed discontinuous Galerkin method for a class of Stokes eigenvalue problems

1.
School of Mathematical Sciences, Guizhou Normal University, Guiyang 550025, Guizhou, China
2.
School of Biology and Engineering, Guizhou Medical University, Guiyang 550025, Guizhou, China

Received: 07 April 2023 Revised: 09 June 2023 Accepted: 25 June 2023 Published: 03 July 2023
MSC : 65N25, 65N30

For a class of Stokes eigenvalue problems including the classical Stokes eigenvalue problem and the magnetohydrodynamic Stokes eigenvalue problem a residual type a posteriori error estimate of the mixed discontinuous Galerkin finite element method using $ \mathbb{P}_{k}-\mathbb{P}_{k-1} $ element $ (k\geq 1) $ is studied in this paper. The a posteriori error estimators for approximate eigenpairs are given. The reliability and efficiency of the posteriori error estimator for the eigenfunction are proved and the reliability of the estimator for the eigenvalue is also analyzed. The numerical results are provided to confirm the theoretical predictions and indicate that the method considered in this paper can reach the optimal convergence order $ O(dof^{\frac{-2k}{d}}) $.
Citation: Lingling Sun, Hai Bi, Yidu Yang. A posteriori error estimates of mixed discontinuous Galerkin method for a class of Stokes eigenvalue problems[J]. AIMS Mathematics, 2023, 8(9): 21270-21297. doi: 10.3934/math.20231084

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Abstract

For a class of Stokes eigenvalue problems including the classical Stokes eigenvalue problem and the magnetohydrodynamic Stokes eigenvalue problem a residual type a posteriori error estimate of the mixed discontinuous Galerkin finite element method using $ \mathbb{P}_{k}-\mathbb{P}_{k-1} $ element $ (k\geq 1) $ is studied in this paper. The a posteriori error estimators for approximate eigenpairs are given. The reliability and efficiency of the posteriori error estimator for the eigenfunction are proved and the reliability of the estimator for the eigenvalue is also analyzed. The numerical results are provided to confirm the theoretical predictions and indicate that the method considered in this paper can reach the optimal convergence order $ O(dof^{\frac{-2k}{d}}) $.

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