The a posteriori error estimates and adaptive computation of nonconforming mixed finite elements for the Stokes eigenvalue problem

Highlights：

• For the Stokes eigenvalue problem, we present a direct analysis, which does not involve the Helmholtz decomposition, for estimating the reliability bound. We give an a posteriori error formula in Rd (d=2,3) which contains the residual term and the consistent error term, and we use two technique lemmas (Lemma 3.1 in [1] for the CR element and Lemma 5.6.4 in [37] for the ECR element) to estimate the consistent term and prove the reliability of the estimator for the problem.

• We give adaptive finite element algorithm based on direct solution (direct-AFEM) and the adaptive finite element algorithm based on the shifted inverse iteration (shifted inverse-AFEM) for the Stokes eigenvalue problem. We provide numerical experiments both in two and three-dimensional domains. Numerical results indicate that the numerical eigenvalues obtained by the above two methods achieve the optimal convergence order O(dof−2/d) and approximate the exact ones from below.

摘要

•For the Stokes eigenvalue problem, we give a new expression for the a priori error estimate of eigenfunction, which shows clearly that ∥uh−u∥0 is a small quantity of higher order compared with ∥uh−u∥h+∥σh−σ∥0.•For the Stokes eigenvalue problem, we present a direct analysis, which does not involve the Helmholtz decomposition, for estimating the reliability bound. We give an a posteriori error formula in Rd (d=2,3) which contains the residual term and the consistent error term, and we use two technique lemmas (Lemma 3.1 in [1] for the CR element and Lemma 5.6.4 in [37] for the ECR element) to estimate the consistent term and prove the reliability of the estimator for the problem.•We give adaptive finite element algorithm based on direct solution (direct-AFEM) and the adaptive finite element algorithm based on the shifted inverse iteration (shifted inverse-AFEM) for the Stokes eigenvalue problem. We provide numerical experiments both in two and three-dimensional domains. Numerical results indicate that the numerical eigenvalues obtained by the above two methods achieve the optimal convergence order O(dof−2/d) and approximate the exact ones from below.