Steklov Eigenvalue Problem Research Articles

Convergence and quasi-optimality of adaptive FEM for Steklov eigenvalue problems

The main goal of this article is the study of the convergence and optimality properties of adaptive finiteelement methods (AFEMs) for Steklov eigenvalue problems. These problems arise, for example, in the analysis of the stability of mechanical oscillators immersed in a viscous fluid (see Conca et al., 1995 and the references therein) or in the study of vibration modes of a structure interacting with an incompressible fluid (Bermudez et al., 2000). Finite-element approximations for these problems have been widely used and analysed under a general framework. AFEMs make efficient use of the computational resources, and for certain problems AFEMs are even indispensable for their numerical resolvability. The ultimate goal of adaptive methods is to equidistribute the error and the computational effort to obtain a sequence of meshes with optimal complexity. Historically, the first step to prove optimality has been to understand the convergence of adaptive methods. A basic result for the convergence of linear problems was presented in Morin et al. (2008), where very general conditions on the linear problems and on the ingredients of the AFEM that guarantee convergence were stated. Following these ideas, the (plain) convergence of an AFEM for elliptic eigenvalue problems was proved in Garau et al. (2009). The optimality of an AFEM using Dorfler’s marking strategy (Dorfler, 1996) was proved by Stevenson (2007) and Cascon et al. (2008) for linear elliptic problems. The linear convergence of an AFEM for elliptic eigenvalue problems was proved in Giani & Graham (2009) when considering the approximation of simple eigenvalues, provided that the initial mesh is sufficiently fine, and an improvement of this AFEM including optimality was presented in Dai et al. (2008). In this article we consider the Steklov eigenvalue problem that consists of finding λ ∈ R and u 6≡ 0 such that −∇ ∙ (A∇u)+ cu = 0 in Ω, (A∇u) ∙ −→n = λρu on ∂Ω,

Constant-sign and sign-changing solutions for nonlinear elliptic equations with Neumann boundary values

In this paper we study the existence of multiple solutions to the equation \begin{align*} -{\Delta_p} u = f(x,u)-|u|^{p-2}u \end{align*} with the nonlinear boundary condition \begin{align*} |\nabla u|^{p-2} \frac{\partial u}{ \partial \nu} = \lambda |u|^{p-2}u+g(x,u). \end{align*} We establish the existence of a smallest positive solution, a greatest negative solution, and a nontrivial sign-changing solution when the parameter $\lambda$ is greater than the second eigenvalue of the Steklov eigenvalue problem. Our approach is based on truncation techniques and comparison principles for nonlinear elliptic differential inequalities. In particular, we make use of variational and topological tools, such as critical point theory, the mountain-pass theorem, the second deformation lemma and variational characterizations of the second eigenvalue of the Steklov eigenvalue problem.

A two-grid discretization scheme for the Steklov eigenvalue problem

In the paper, a two-grid discretization scheme is discussed for the Steklov eigenvalue problem. With the scheme, the solution of the Steklov eigenvalue problem on a fine grid is reduced to the solution of the Steklov eigenvalue problem on a much coarser grid and the solution of a linear algebraic system on the fine grid. Using spectral approximation theory, it is shown theoretically that the two-scale scheme is efficient and the approximate solution obtained by the scheme maintains the asymptotically optimal accuracy. Finally, numerical experiments are carried out to confirm the considered theory.

Simulation of a nonlinear Steklov eigenvalue problem using finite-element approximation

Elliptic problems with parameters in the boundary conditions are called Steklov problems. With the tool of computational approximation (finite-element method), we estimate the solution of a nonlinear Steklov eigenvalue problem for a second-order, self-adjoint, elliptic differential problem. We discussed the behavior of the nonlinear problem with the help of computational results using Matlab.

Nonconforming finite element approximations of the Steklov eigenvalue problem

This paper deals with nonconforming finite element approximations of the Steklov eigenvalue problem. For a class of nonconforming finite elements, it is shown that the j-th approximate eigenpair converges to the j-th exact eigenpair and error estimates for eigenvalues and eigenfunctions are derived. Furthermore, it is proved that the j-th eigenvalue derived by the EQ 1 rot element gives lower bound of the j-th exact eigenvalue, whereas the nonconforming Crouzeix–Raviart element and the Q 1 rot element provide lower bounds of the large eigenvalues. Numerical results are presented to confirm the considered theory.

Extrapolation and superconvergence of the Steklov eigenvalue problem

On the basis of a transform lemma, an asymptotic expansion of the bilinear finite element is derived over graded meshes for the Steklov eigenvalue problem, such that the Richardson extrapolation can be applied to increase the accuracy of the approximation, from which the approximation of O(h 3.5) is obtained. In addition, by means of the Rayleigh quotient acceleration technique and an interpolation postprocessing method, the superconvergence of the bilinear finite element is presented over graded meshes for the Steklov eigenvalue problem, and the approximation of O(h 3) is gained. Finally, numerical experiments are provided to demonstrate the theoretical results.

Eigenvalues of the [formula omitted]-Laplacian Steklov problem

Consider Steklov eigenvalue problem involving the p ( x ) -Laplacian on a bounded domain Ω, the open subset of R N with N ⩾ 2 , as follows { Δ p ( x ) u = | u | p ( x ) − 2 u in Ω , | ∇ u | p ( x ) − 2 ∂ u ∂ γ = λ | u | p ( x ) − 2 u on ∂ Ω , where p ( x ) ≢ constant. We prove that the existence of infinitely many eigenvalue sequences. Unlike the p-Laplacian case, for a variable exponent p ( x ) (≢ constant), there does not exist a principal eigenvalue and the set of all eigenvalues is not closed under some assumptions. Finally, we present some sufficient conditions for the infimum of all eigenvalues is zero and positive, respectively.

A posteriori error estimates for the Steklov eigenvalue problem

In this paper we introduce and analyze an a posteriori error estimator for the linear finite element approximations of the Steklov eigenvalue problem. We define an error estimator of the residual type which can be computed locally from the approximate eigenpair and we prove that, up to higher order terms, the estimator is equivalent to the energy norm of the error. Finally, we prove that the volumetric part of the residual term is dominated by a constant times the edge residuals, again up to higher order terms.

Steklov eigenvalues for the $\infty$-Laplacian

We study the Steklov eigenvalue problem for the \infty -laplacian. To this end we consider the limit as p \to \infty of solutions of -\Delta_p u_p =0 in a domain \Omega with |\nabla u_p|^{p-2} \partial u_p / \partial \nu = \lambda |u|^{p-2} u on \partial\Omega . We obtain a limit problem that is satisfied in the viscosity sense and a geometric characterization of the second eigenvalue.

Visual Acuity

In this paper, by explicitly calculating the principal symbols of pseudodifferential operators, we establish two Weyl-type asymptotic formulas with sharp remainder estimates for the counting functions of the two classes of biharmonic Steklov eigenvalue problems of smooth bounded domains in a Riemannian manifold.

On a fourth order Steklov eigenvalue problem

SummaryWe study the spectrum of a biharmonic Steklov eigenvalue problem in a bounded domain of R

The effect of reduced integration in the Steklov eigenvalue problem

In this paper we analyze the effect of introducing a numerical integration in the piecewise linear finite element approximation of the Steklov eigenvalue problem. We obtain optimal order error estimates for the eigenfunctions when this numerical integration is used and we prove that, for singular eigenfunctions, the eigenvalues obtained using this reduced integration are better approximations than those obtained using exact integration when the mesh size is small enough.

The best constant in a trace inequality

The best possible constant in the trace inequality ∥v∥ L 1(Γ 1) ⩽ 0 · ∥▽v∥ 2(Ω) , ∀v ϵ H 1 Γ 2 (Ω) is shown to be given by a quantity in terms of the solution of an elliptic boundary value problem, where, Γ 1, Γ 2 are disjoint subsets of ∂Ω with positive measures, and the Sobolev space H 1 Γ 2 (Ω) = {v ϵ H 1 (Ω) ¦v = 0 on Γ 2} . Some examples are presented with either an analytic expression or a numerical value for the best constant. As an application, the best constant c 0 is used to give an upper bound for the first eigenvalue of a modified Steklov eigenvalue problem.