17 research outputs found
Fully computable a posteriori error bounds for eigenfunctions
Fully computable a posteriori error estimates for eigenfunctions of compact
self-adjoint operators in Hilbert spaces are derived. The problem of
ill-conditioning of eigenfunctions in case of tight clusters and multiple
eigenvalues is solved by estimating the directed distance between the spaces of
exact and approximate eigenfunctions. Derived upper bounds apply to various
types of eigenvalue problems, e.g. to the (generalized) matrix, Laplace, and
Steklov eigenvalue problems. These bounds are suitable for arbitrary conforming
approximations of eigenfunctions, and they are fully computable in terms of
approximate eigenfunctions and two-sided bounds of eigenvalues. Numerical
examples illustrate the efficiency of the derived error bounds for
eigenfunctions.Comment: 27 pages, 8 tables, 9 figure
Guaranteed Lower Eigenvalue Bound of Steklov Operator with Conforming Finite Element Methods
For the eigenvalue problem of the Steklov differential operator, by following
Liu's approach, an algorithm utilizing the conforming finite element method
(FEM) is proposed to provide guaranteed lower bounds for the eigenvalues. The
proposed method requires the a priori error estimation for FEM solution to
nonhomogeneous Neumann problems, which is solved by constructing the
hypercircle for the corresponding FEM spaces and boundary conditions. Numerical
examples are also shown to confirm the efficiency of our proposed method.Comment: 21 pages, 4 figures, 4 table