scholarly journals Local defect-correction method based on multilevel discretization for Steklov eigenvalue problem

2021 ◽  
Author(s):  
Fei Xu ◽  
Liu Chen ◽  
Qiumei Huang
Keyword(s):  
Eigenvalue Problem ◽  
Large Scale ◽  
Eigenvalue Problems ◽  
Correction Method ◽  
Small Scale ◽  
Local Defect ◽  
Defect Correction ◽  
Steklov Eigenvalue ◽  
Steklov Eigenvalue Problem ◽  
Steklov Eigenvalue Problems

In this paper, we propose a local defect-correction method for solving the Steklov eigenvalue problem arising from the scalar second order positive definite partial differential equations based on the multilevel discretization. The objective is to avoid solving large-scale equations especially the large-scale Steklov eigenvalue problem whose computational cost increases exponentially. The proposed algorithm transforms the Steklov eigenvalue problem into a series of linear boundary value problems, which are defined in a multigrid space sequence, and a series of small-scale Steklov eigenvalue problems in a coarse correction space. Furthermore, we use the local defect-correction technique to divide the large-scale boundary value problems into small-scale subproblems. Through our proposed algorithm, we avoid solving large-scale Steklov eigenvalue problems. As a result, our proposed algorithm demonstrates significantly improved the solving efficiency. Additionally, we conduct numerical experiments and a rigorous theoretical analysis to verify the effectiveness of our proposed approach.

scholarly journals Fractional eigenvalue problems that approximate Steklov eigenvalue problems

2017 ◽  
Vol 148 (3) ◽  
pp. 499-516 ◽  
Author(s):  
Leandro M. Del Pezzo ◽  
Julio D. Rossi ◽  
Ariel M. Salort
Keyword(s):  
Eigenvalue Problem ◽  
Eigenvalue Problems ◽  
Steklov Eigenvalue ◽  
Steklov Eigenvalue Problem ◽  
Non Local ◽  
Steklov Eigenvalue Problems ◽  
Fractional Type

In this paper we analyse possible extensions of the classical Steklov eigenvalue problem to the fractional setting. In particular, we find a non-local eigenvalue problem of fractional type that approximates, when taking a suitable limit, the classical Steklov eigenvalue problem.

Optimal Estimates for Steklov Eigenvalue Gaps and Ratios on Warped Product Manifolds

2019 ◽  
Author(s):  
Changwei Xiong
Keyword(s):  
Riemannian Manifold ◽  
Warped Product ◽  
Eigenvalue Problems ◽  
Euclidean Ball ◽  
Convex Boundary ◽  
Steklov Eigenvalue ◽  
Optimal Estimates ◽  
Steklov Eigenvalue Problem ◽  
Steklov Eigenvalue Problems ◽  
Optimal Lower Bound

Abstract We consider an $n$-dimensional smooth Riemannian manifold $M^n=[0,R)\times \mathbb{S}^{n-1}$ endowed with a warped product metric $g=dr^2+h^2(r)g_{\mathbb{S}^{n-1}}$ and diffeomorphic to a Euclidean ball. Suppose that $M$ has strictly convex boundary. First, for the classical Steklov eigenvalue problem, we derive an optimal lower (upper, respectively) bound for its eigenvalue gaps in terms of $h^{\prime}(R)/h(R)$ when $n\geq 2$ and $Ric_g\geq 0$ ($\leq 0$, respectively). Second, in the same spirit, for two 4th-order Steklov eigenvalue problems studied by Kuttler and Sigillito in 1968, we deduce an optimal lower bound for their eigenvalue gaps in terms of either $h^{\prime}(R)/h^3(R)$ or $h^{\prime}(R)/h(R)$ when $n=2$ and the Gaussian curvature is nonnegative. We also consider optimal estimates on the eigenvalue ratios for these eigenvalue problems.

A Multilevel Correction Method for Steklov Eigenvalue Problem by Nonconforming Finite Element Methods

2015 ◽  
Vol 8 (3) ◽  
pp. 383-405 ◽  
Author(s):  
Xiaole Han ◽  
Yu Li ◽  
Hehu Xie
Keyword(s):  
Finite Element ◽  
Eigenvalue Problem ◽  
Finite Element Methods ◽  
Correction Method ◽  
Nonconforming Finite Element ◽  
Element Space ◽  
Correction Scheme ◽  
Steklov Eigenvalue ◽  
Steklov Eigenvalue Problem ◽  
Nonconforming Finite Element Methods

AbstractIn this paper, a multilevel correction scheme is proposed to solve the Steklov eigenvalue problem by nonconforming finite element methods. With this new scheme, the accuracy of eigenpair approximations can be improved after each correction step which only needs to solve a source problem on finer finite element space and an Steklov eigenvalue problem on the coarsest finite element space. This correction scheme can increase the overall efficiency of solving eigenvalue problems by the nonconforming finite element method. Furthermore, as same as the direct eigenvalue solving by nonconforming finite element methods, this multilevel correction method can also produce the lower-bound approximations of the eigenvalues.

scholarly journals Computation of free boundary minimal surfaces via extremal Steklov eigenvalue problems

2021 ◽  
Author(s):  
Edouard Oudet ◽  
Chiu-Yen Kao ◽  
Braxton Osting
Keyword(s):  
Numerical Methods ◽  
Free Boundary ◽  
Minimal Surfaces ◽  
Mean Curvature ◽  
Eigenvalue Problems ◽  
Compact Surface ◽  
Steklov Eigenvalue ◽  
Steklov Eigenvalue Problem ◽  
Zero Mean Curvature ◽  
Steklov Eigenvalue Problems

Recently Fraser and Schoen showed that the solution of a certain extremal Steklov eigenvalue problem on a compact surface with boundary can be used to generate a free boundary minimal surface, i.e., a surface contained in the ball that has (i) zero mean curvature and (ii) meets the boundary of the ball orthogonally. In this paper, we develop numerical methods that use this connection to realize free boundary minimal surfaces. Namely, on a compact surface, Σ, with genus γ and b boundary components, we maximize σ j (Σ, g) L(∂Σ, g) over a class of smooth metrics, g, where σ j (Σ, g) is the j-th nonzero Steklov eigenvalue and L(∂Σ, g) is the length of ∂Σ.

scholarly journals A Prediction-Correction Dynamic Method for Large-Scale Generalized Eigenvalue Problems

2013 ◽  
Vol 2013 ◽  
pp. 1-8
Author(s):  
Xin-long Luo ◽  
Jia-ru Lin ◽  
Wei-ling Wu
Keyword(s):  
Dynamical System ◽  
Eigenvalue Problem ◽  
Large Scale ◽  
Eigenvalue Problems ◽  
Correction Method ◽  
Differential Algebraic Equations ◽  
Euler Method ◽  
Generalized Eigenvalue Problem ◽  
Algebraic Equations ◽  
Generalized Eigenvalue

This paper gives a new prediction-correction method based on the dynamical system of differential-algebraic equations for the smallest generalized eigenvalue problem. First, the smallest generalized eigenvalue problem is converted into an equivalent-constrained optimization problem. Second, according to the Karush-Kuhn-Tucker conditions of this special equality-constrained problem, a special continuous dynamical system of differential-algebraic equations is obtained. Third, based on the implicit Euler method and an analogous trust-region technique, a prediction-correction method is constructed to follow this system of differential-algebraic equations to compute its steady-state solution. Consequently, the smallest generalized eigenvalue of the original problem is obtained. The local superlinear convergence property for this new algorithm is also established. Finally, in comparison with other methods, some promising numerical experiments are presented.

Steklov eigenvalue problem with a-harmonic solutions and variable exponents

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Belhadj Karim ◽  
Abdellah Zerouali ◽  
Omar Chakrone
Keyword(s):  
Eigenvalue Problem ◽  
Smooth Boundary ◽  
Normal Vector ◽  
Variable Exponents ◽  
Variational Technique ◽  
Unit Normal Vector ◽  
Steklov Eigenvalue ◽  
Steklov Eigenvalue Problem ◽  
Outward Unit ◽  
Outward Unit Normal Vector

AbstractUsing the Ljusternik–Schnirelmann principle and a new variational technique, we prove that the following Steklov eigenvalue problem has infinitely many positive eigenvalue sequences:\left\{\begin{aligned} &\displaystyle\operatorname{div}(a(x,\nabla u))=0&&% \displaystyle\phantom{}\text{in }\Omega,\\ &\displaystyle a(x,\nabla u)\cdot\nu=\lambda m(x)|u|^{p(x)-2}u&&\displaystyle% \phantom{}\text{on }\partial\Omega,\end{aligned}\right.where {\Omega\subset\mathbb{R}^{N}}{(N\geq 2)} is a bounded domain of smooth boundary {\partial\Omega} and ν is the outward unit normal vector on {\partial\Omega}. The functions {m\in L^{\infty}(\partial\Omega)}, {p\colon\overline{\Omega}\mapsto\mathbb{R}} and {a\colon\overline{\Omega}\times\mathbb{R}^{N}\mapsto\mathbb{R}^{N}} satisfy appropriate conditions.

Further analysis of the local defect correction method

Computing ◽  
1996 ◽  
Vol 56 (2) ◽  
pp. 117-139 ◽  
Author(s):  
P. J. J. Ferket ◽  
A. A. Reusken
Keyword(s):  
Correction Method ◽  
Local Defect ◽  
Defect Correction

scholarly journals The Steklov Problem on Differential Forms

2019 ◽  
Vol 71 (2) ◽  
pp. 417-435
Author(s):  
Mikhail A. Karpukhin
Keyword(s):  
Eigenvalue Problem ◽  
Spectral Properties ◽  
Differential Forms ◽  
Type Inequality ◽  
Slight Modification ◽  
Dimensional Manifold ◽  
Hodge Laplacian ◽  
Steklov Eigenvalue ◽  
Steklov Eigenvalue Problem ◽  
Dirichlet To Neumann Map

AbstractIn this paper we study spectral properties of the Dirichlet-to-Neumann map on differential forms obtained by a slight modification of the definition due to Belishev and Sharafutdinov. The resulting operator $\unicode[STIX]{x039B}$ is shown to be self-adjoint on the subspace of coclosed forms and to have purely discrete spectrum there. We investigate properties of eigenvalues of $\unicode[STIX]{x039B}$ and prove a Hersch–Payne–Schiffer type inequality relating products of those eigenvalues to eigenvalues of the Hodge Laplacian on the boundary. Moreover, non-trivial eigenvalues of $\unicode[STIX]{x039B}$ are always at least as large as eigenvalues of the Dirichlet-to-Neumann map defined by Raulot and Savo. Finally, we remark that a particular case of $p$-forms on the boundary of a $2p+2$-dimensional manifold shares many important properties with the classical Steklov eigenvalue problem on surfaces.

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