Chun'guang You

Chun'guang You, Hehu Xie, Xuefeng Liu

To provide mathematically rigorous eigenvalue bounds for the Steklov eigenvalue problem, an enhanced version of the eigenvalue estimation algorithm developed by the third author is proposed, which removes the requirements of the positive definiteness of bilinear forms in the formulation of eigenvalue problems. In practical eigenvalue estimation, the Crouzeix--Raviart finite element method (FEM) along with quantitative error estimation is adopted. Numerical experiments for eigenvalue problems defined on a square domain and an L-shaped domain are provided to validate the precision of computed eigenvalue bounds.

Xuefeng Liu, Chun'guang You

For the quadratic Lagrange interpolation function, an algorithm is proposed to provide explicit and verified bound for the interpolation error constant that appears in the interpolation error estimation. The upper bound for the interpolation constant is obtained by solving an eigenvalue problem along with explicit lower bound for its eigenvalues. The lower bound for interpolation constant can be easily obtained by applying the Rayleigh-Ritz method. Numerical computation is performed to demonstrate the sharpness of lower and upper bounds of the interpolation constants over triangles of different shapes. An online computing demo is available at http://www.xfliu.org/onlinelab/.