Numerical methods for spectral problems: theory and applications
Guaranteed bounds for the eigenvalues of Laplacian in curved domains
*Shinya Uchiumi (Gakushuin University)
We consider the eigenvalue problem of the Laplacian with the homogeneous Dirichlet boundary condition in curved domains. Based on the theorem by Liu and Oishi , we show guaranteed bounds of the eigenvalues, where the lower and upper bounds are explicitly computable. This talk includes two main discussions. The first one is construction of a finite dimensional eigenvalue problem. We construct a curved finite element space using the mapping by Zlamal . The second discussion is an estimate of the constant that appears in the theorem by Liu and Oishi. The construction of the finite dimensional problem and computation of the constants are performed with kv library , which deals with the interval arithmetic and the power series arithmetic.
Math formula preview:
Eigenvalue problem, curved domain, finite element method