417 research outputs found
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
Steklov problem on differential forms
In this paper we study spectral properties of Dirichlet-to-Neumann map on
differential forms obtained by a slight modification of the definition due to
Belishev and Sharafutdinov. The resulting operator is shown to be
self-adjoint on the subspace of coclosed forms and to have purely discrete
spectrum there.We investigate properies of eigenvalues of and prove a
Hersch-Payne-Schiffer type inequality relating products of those eigenvalues to
eigenvalues of Hodge Laplacian on the boundary. Moreover, non-trivial
eigenvalues of are always at least as large as eigenvalues of
Dirichlet-to-Neumann map defined by Raulot and Savo. Finally, we remark that a
particular case of -forms on the boundary of -dimensional manifold
shares a lot of important properties with the classical Steklov eigenvalue
problem on surfaces.Comment: 18 page
Exact constants in Poincare type inequalities for functions with zero mean boundary traces
In the paper, we investigate Poincare type inequalities for the functions
having zero mean value on the whole boundary of a Lipschitz domain or on a
measurable part of the boundary. We derive exact and easily computable
constants for some basic domains (rectangles, cubes, and right triangles). In
the last section, we derive an a estimate of the difference between the exact
solutions of two boundary value problems. Constants in Poincare type
inequalities enter these estimates, which provide guaranteed a posteriori error
control.Comment: A gap in the proof of Theorem 3.2 is fixed; 19 pages, 3 figure
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