Framework for verified lower eigenvalue bound for self-adjoint differential operators

摘要

A universal framework is proposed to give explicit lower and upper bounds for the eigenvalues of self-adjoint differential operators. In the case of the Laplacian operator, by applying CrouzeixRaviart finite elements, an efficient algorithm is developed to bound the eigenvalues for the Laplacian defined in 1D, 2D and 3D spaces. For biharmonic operators, Fujino-Morley FEM is adopted to bound the eigenvalues. By further adopting the interval arithmetic, the explicit eigenvalue bounds from numerical computations can be mathematically correct.