Lower spectral bounds by Wilson's brick discretization

This paper discusses the Wilson element approximation for the eigenvalue problem of Laplace operator on n-dimensional polygonal domain (n=2,3), and the main results are as follows: (1) We establish the relationship between the interpolation weak estimate of the Wilson element and the interpolation weak estimate of n-linear element. (2) We prove that 3-dimensional Wilson's brick eigenvalues approximate the exact eigenvalues from below, and thereby make a new progress on such an open problem in the finite element method.

[1]  Jun Hu,et al.  THE LOWER APPROXIMATION OF EIGENVALUE BY LUMPED MASS FINITE ELEMENT METHOD ∗1) , 2004 .

[2]  M. Zlámal,et al.  Convergence of the nonconforming Wilson element for arbitrary quadrilateral meshes , 1980 .

[3]  Zhimin Zhang,et al.  Analysis of Some Quadrilateral Nonconforming Elements for Incompressible Elasticity , 1997 .

[4]  Zhimin Zhang,et al.  Wilson's element for the Reissner-Mindlin plate , 1994 .

[5]  Qun Lin,et al.  New expansions of numerical eigenvalues by Wilson's element , 2009 .

[6]  Yidu Yang,et al.  Nonconforming finite element approximations of the Steklov eigenvalue problem , 2009 .

[7]  G. Strang,et al.  An Analysis of the Finite Element Method , 1974 .

[8]  Bo Li,et al.  Superconvergence analysis and error expansion for the Wilson nonconforming finite element , 1994 .

[9]  Philippe G. Ciarlet,et al.  The finite element method for elliptic problems , 2002, Classics in applied mathematics.

[10]  Yidu Yang,et al.  TWO-GRID DISCRETIZATION SCHEMES OF THE NONCONFORMING FEM FOR EIGENVALUE PROBLEMS * , 2009 .

[11]  Zhimin Zhang,et al.  Eigenvalue approximation from below using non-conforming finite elements , 2010 .

[12]  S. Fenves.  Numerical and computer methods in structural mechanics , 1973 .

[13]  Zhongci Shi.  A convergence condition for the quadrilateral Wilson element , 1984 .

[14]  Yang,et al.  A POSTERIORI ERROR ESTIMATES IN ADINI FINITE ELEMENT FOR EIGENVALUE PROBLEMS , 2000 .

[15]  Qun Lin,et al.  New expansions of numerical eigenvalues for -Δu = λρu by nonconforming elements , 2008, Math. Comput..

[16]  Zhong-Ci Shi,et al.  A new superconvergence property of Wilson nonconforming finite element , 1997 .

[17]  Yidu Yang,et al.  The order-preserving convergence for spectral approximation of self-adjoint completely continuous operators , 2008 .