Computation of tight enclosures for Laplacian eigenvalues

Séminaire CRM-CAMP Spotlight on Graduate Research (3 déc. 2020 / Dec. 3, 2020)

Joel Dahne (Uppsala University, Sweden)

Abstract: Recently, there has been interest in high-precision approximations of the fundamental eigenvalue of the Laplace-Beltrami operator on spherical triangles for combinatorial purposes. We present computations of improved and rigorous enclosures to these eigenvalues. This is achieved by applying the method of particular solutions in high precision, the enclosure being obtained by a combination of interval arithmetic and Taylor models. The index of the eigenvalue can be certified by exploiting the monotonicity of the eigenvalue with respect to the domain. The classically troublesome case of singular corners we handle by combining expansions from all singular corners and an expansion from an interior point.