Finite Element Methods For Steklov Type Eigenvalue Problems

In this paper,the finite element methods for solving Steklov type eigenvalue problems are studied,including: the Steklov eigenvalue problem in inverse scattering,the Steklov eigenvalue problem in linear elasticity,the Steklov eigenvalue problem in fluid-solid vibration problem,and modified interior transmission eigenvalue problem.Firstly,the discontinuous Galerkin(DG)finite element method for Steklov eigenvalue problems in inverse scattering is studied.A complete error estimation including a improved prior error estimation and a posterior error estimation are discussed.At the same time,the demonstration of the posterior error estimator of the eigenfunction is reliable and effective by using the enrichment operator and the lifting operator,the reliability of the estimator for the eigenvalue error estimation are analyzed.In two and three dimension domains,the numerical experiments in the adaptive fashion can reach the optimal convergence order,which is coincide with the theoretical analysis.Secondly,DG finite element method of Nitsche’s version for the linear elastic Steklov-Lam′e eigenvalue problem is studied.The a priori error estimates independent of Lam′e parameters are derived under the condition of low regularity,and DG method is proved to be lockingfree for this problem.The Lam′e parameters are changed in different domains for numerical experiments to verify the effectiveness and robustness of the proposed method.Thirdly,the nonconforming Crouzeix-Raviart(CR)finite element method for fluid-solid vibration Steklov eigenvalue problem is studied.The a priori error estimates are derived by combining the error estimation of the source problem and the operator convergence in the low regularity,and the reliability and efficiency of the residual type posterior error estimator are proved.Both theoretical analysis and numerical experiments show that the CR eigenvalue is a lower bound of the exact eigenvalue.Finally,a conforming finite element method for the modified interior transmission eigenvalue problem in inverse scattering is studied.Gintides described the problem as a new Steklov eigenvalue problem.Under the assumption of low regularity of a completely reasonable solution,the problem is analyzed and proved completely by using the G?arding inequality and the T-coercive technique in metamaterial and natural cases.Both theoretical analysis and numerical results show that the posterior error estimators are reliable and effective.The experiments also show that the conforming element eigenvalue approximates the exact eigenvalue from below,and the approximation can reach the optimal convergence order.