Lower Eigenvalue Bound Formula

Usuario

Show the formula to calculate lower eigenvalue bound for Laplacian using Crouzeix-Raviart FEM.

Guaranteed computation on eigenvalue problems

The formula to calculate the lower eigenvalue bound for the Laplacian using Crouzeix-Raviart Finite Element Method (FEM) is given as follows:

[ \lambda_k \geq \lambda_k := \lambda_{kh} (1 + 0.18932 h^2 \lambda_{kh}) ]

Here, ( \lambda_k ) represents the ( k )-th eigenvalue, ( \lambda_{kh} ) is the ( k )-th approximate eigenvalue obtained from the Crouzeix-Raviart FEM, and ( h ) is the mesh size【8†source】.

The formula to calculate the lower eigenvalue bound for the Laplacian using Crouzeix-Raviart Finite Element Method (FEM) is given as follows: \[ \lambda_k \geq \lambda_k := \lambda_{kh} (1 + 0.18932 h^2 \lambda_{kh}) \] Here, \( \lambda_k \) represents the \( k \)-th eigenvalue, \( \lambda_{kh} \) is the \( k \)-th approximate eigenvalue obtained from the Crouzeix-Raviart FEM, and \( h \) is the mesh size【8†source】.