Local a priori/a posteriori error estimates of conforming finite elements approximation for Steklov eigenvalue problems

Based on the work of Xu and Zhou(2000),this paper makes a further discussion on conforming finite elements approximation for Steklov eigenvalue problems,and proves a local a priori error estimate and a new local a posteriori error estimate in ||·||1,Ω0 norm for conforming elements eigenfunction,which has not been studied in existing literatures.

A family of symmetric mixed finite elements for linear elasticity on tetrahedral grids

A family of stable mixed finite elements for the linear elasticity on tetrahedral grids are constructed,where the stress is approximated by symmetric H(div)-Pk polynomial tensors and the displacement is approximated by C-1-Pk-1polynomial vectors,for all k 4.The main ingredients for the analysis are a new basis of the space of symmetric matrices,an intrinsic H(div)bubble function space on each element,and a new technique for establishing the discrete inf-sup condition.In particular,they enable us to prove that the divergence space of the H(div)bubble function space is identical to the orthogonal complement space of the rigid motion space with respect to the vector-valued Pk-1polynomial space on each tetrahedron.The optimal error estimate is proved,verified by numerical examples.

FINITE ELEMENT METHODS FOR A BI-WAVE EQUATION MODELING D-WAVE SUPERCONDUCTORS

In this paper we develop two conforming finite element methods for a fourth order bi-wave equation arising as a simplified Ginzburg-Landau-type model for d-wave superconductors in absence of applied magnetic field. Unlike the biharmonic operator A2, the bi-wave operator □^2 is not an elliptic operator, so the energy space for the bi-wave equation is much larger than the energy space for the biharmonic equation. This then makes it possible to construct low order conforming finite elements for the bi-wave equation. However, the existence and construction of such finite elements strongly depends on the mesh. In the paper, we first characterize mesh conditions which allow and not allow construction of low order conforming finite elements for approximating the bi-wave equation. We then construct a cubic and a quartic conforming finite element. It is proved that both elements have the desired approximation properties, and give optimal order error estimates in the energy norm, suboptimal （and optimal in some cases） order error estimates in the H1 and L^2 norm. Finally, numerical experiments are presented to guage the efficiency of the proposed finite element methods and to validate the theoretical error bounds.

NEW ERROR ESTIMATES FOR LINEAR TRIANGLE FINITE ELEMENTS IN THE STEKLOV EIGENVALUE PROBLEM

This paper is concerned with the finite elements approximation for the Steklov eigen- value problem on concave polygonal domain. We make full use of the regularity estimate and the characteristic of edge average interpolation operator of nonconforming Crouzeix- Raviart element, and prove a new and optimal error estimate in ｜｜ ｜｜o,δΩ for the eigenfunc- tion of linear conforming finite element and the nonconforming Crouzeix-Raviart element. Finally, we present some numerical results to support the theoretical analysis.

A family of mixed finite elements for the biharmonic equations on triangular and tetrahedral grids

This paper introduces a new family of mixed finite elements for solving a mixed formulation of the biharmonic equations in two and three dimensions.The symmetric stress σ=−∇^(2)u is sought in the Sobolev space H(divdiv,Ω;S)simultaneously with the displacement u in L^(2)(Ω).By stemming from the structure of H(div,Ω;S)conforming elements for the linear elasticity problems proposed by Hu and Zhang(2014),the H(divdiv,Ω;S)conforming finite element spaces are constructed by imposing the normal continuity of divσ on the H(div,Ω;S)conforming spaces of P_(k) symmetric tensors.The inheritance makes the basis functions easy to compute.The discrete spaces for u are composed of the piecewise P_(k−2) polynomials without requiring any continuity.Such mixed finite elements are inf-sup stable on both triangular and tetrahedral grids for k≥3,and the optimal order of convergence is achieved.Besides,the superconvergence and the postprocessing results are displayed.Some numerical experiments are provided to demonstrate the theoretical analysis.

Upper spectral bounds and a posteriori error analysis of several mixed finite element approximations for the Stokes eigenvalue problem

This paper discusses conforming mixed finite element approximations for the Stokes eigenvalue problem. Firstly, several mixed finite element identities are proved. Based on these identities, the following new results are given： （1） It is proved that the numerical eigenvalues obtained by mini-element, P1-P1 element and Q1-Q1 element approximate the exact eigenvalues from above. （2） As for the P1-P1, Q1-Q1 and Q1-Po element eigenvalues, the asymptotically exact a posteriori error indicators are presented. （3） The reliable and efficient a posteriori error estimator proposed by Verfiirth is applied to mini-element eigenfunctions. Finally, numerical experiments are carried out to verify the theoretical analysis.