Elliptic Eigenvalue Problems Research Articles

Two-grid methods for a class of nonlinear elliptic eigenvalue problems

In this article, we introduce and analyse some two-grid methods for nonlinear elliptic eigenvalue problems of the form div(Du)+Vu+f(u2)u=»u,%u%L2=1. We provide a priori error estimates for the ground state energy, the eigenvalue » and the eigenfunction u, in various Sobolev norms. We focus in particular on the Fourier spectral approximation (for periodic boundary conditions), and on the P1 and P2 finite element discretizations (for homogeneous Dirichlet boundary conditions), taking numerical integration errors into account. Finally, we provide numerical examples illustrating our analysis.

Superconvergence two-grid scheme based on shifted-inverse power method for eigenvalue problems by function value recovery

In the paper, an improved two-grid scheme based on shifted-inverse power method is proposed to solve the elliptic eigenvalue problems. With this new scheme, the solution of the elliptic eigenvalue problem on a fine grid Th is reduced to the solution of the elliptic eigenvalue problem and the recovered eigenfunction on a much coarser grid TH, and the solution of an elliptic boundary value problem and the recovered solution on the fine grid Th. Theoretical analysis shows that the scheme allows a much coarser mesh to achieve the superconvergence rate. Finally, some numerical experiments are carried out to confirm the theoretical analysis.

A reaction–diffusion model of harmful algae and zooplankton in an ecosystem

This paper is devoted to the investigation of an unstirred chemostat system modeling the interactions of two essential nutrients (i.e., nitrogen and phosphorus), harmful algae (i.e., P. parvum and cyanobacteria), and a small-bodied zooplankton in an ecosystem. To obtain a weakly repelling property of a compact and invariant set on the boundary, we introduce an associated elliptic eigenvalue problem. It turns out that the model system admits a coexistence steady state and is uniformly persistent provided that the trivial steady state, two semi-trivial steady states and a global attractor on the boundary are all weak repellers.

Simultaneous reduced basis approximation of parameterized elliptic eigenvalue problems

The focus is on a model reduction framework for parameterized elliptic eigenvalue problems by a reduced basis method. In contrast to the standard single output case, one is interested in approximating several outputs simultaneously, namely a certain number of the smallest eigenvalues. For a fast and reliable evaluation of these input-output relations, we analyze a posteriori error estimators for eigenvalues. Moreover, we present different greedy strategies and study systematically their performance. Special attention needs to be paid to multiple eigenvalues whose appearance is parameter-dependent. Our methods are of particular interest for applications in vibro-acoustics.

Convergence and quasi-optimality of adaptive finite element methods for harmonic forms

Numerical computation of harmonic forms (typically called harmonic fields in three space dimensions) arises in various areas, including computer graphics and computational electromagnetics. The finite element exterior calculus framework also relies extensively on accurate computation of harmonic forms. In this work we study the convergence properties of adaptive finite element methods (AFEM) for computing harmonic forms. We show that a properly defined AFEM is contractive and achieves optimal convergence rate beginning from any initial conforming mesh. This result is contrasted with related AFEM convergence results for elliptic eigenvalue problems, where the initial mesh must be sufficiently fine in order for AFEM to achieve any provable convergence rate.

Stability results for discontinuous nonlinear elliptic and parabolic problems with a S-shaped bifurcation branch of stationary solutions

We study stability of the nonnegative solutions of a discontinuous elliptic eigenvalue problem relevant in several applications as for instance in climate modeling. After giving the explicit expresion of the S-shaped bifurcation diagram $\left( \lambda ,{{\left\| {{\mu }_{\lambda }} \right\|}_{\infty }} \right)$ we show the instability of the decreasing part of the bifurcation curve and the stability of the increasing part. This extends to the case of non-smooth nonlinear terms the well known 1971 result by M.G. Crandall and P.H. Rabinowitz concerning differentiable nonlinear terms. We point out that, in general, there is a lacking of uniquenees of solutions for the associated parabolic problem. Nevertheless, for nondegenerate solutions (crossing the discontinuity value of u in a transversal way) the comparison principle and the uniqueness of solutions hold. The instability is obtained trough a linearization process leading to an eigenvalue problem in which a Dirac delta distribution appears as a coefficient of the differential operator. The stability proof uses a suitable change of variables, the continuuity of the bifurcation branch and the comparison principle for nondegenerate solutions of the parabolic problem.

Local Structure of Singular Profiles for a Derivative Nonlinear Schrödinger Equation

The derivative nonlinear Schrödinger equation is an $L^2$-critical nonlinear dispersive model for Alfvén waves in a long-wavelength asymptotic regime. Recent numerical studies [X. Liu, G. Simpson, and C. Sulem, Phys. D, 262 (2013), pp. 48--58] on an $L^2$-supercritical extension of this equation provide evidence of finite time singularities. Near the singular point, the solution is described by a universal profile that solves a nonlinear elliptic eigenvalue problem depending only on the strength of the nonlinearity. In the present work, we describe the deformation of the profile and its parameters near criticality, combining asymptotic analysis and numerical simulations.

A nonlocal diffusion model with free boundaries in spatial heterogeneous environment

This paper is concerned with the nonlocal diffusive model with double free boundaries in spatial heterogeneous environment, where the spatial heterogeneity is described by the sign indefinite coefficients. Such a model can be used to illustrate the spreading or vanishing of a new or invasive species. Due to the lack of comparison principle in the nonlocal reaction–diffusion equation, many classical methods cannot be used directly to this nonlocal problem. This motivates us to find new techniques. We first establish the spreading–vanishing dichotomy as well as some criteria that ensure the species spreading or vanishing by principal eigenvalues of associated scalar elliptic eigenvalue problems. And then we determine the spreading speed when spreading occurs.

Reduced basis approximation anda posteriorierror estimates for parametrized elliptic eigenvalue problems

We develop a new reduced basis (RB) method for the rapid and reliable approximation of parametrized elliptic eigenvalue problems. The method hinges upon dual weighted residual type a posteriori error indicators which estimate, for any value of the parameters, the error between the high-fidelity finite element approximation of the first eigenpair and the corresponding reduced basis approximation. The proposed error estimators are exploited not only to certify the RB approximation with respect to the high-fidelity one, but also to set up a greedy algorithm for the offline construction of a reduced basis space. Several numerical experiments show the overall validity of the proposed RB approach.

Immersed finite element method for eigenvalue problem

We consider the approximation of elliptic eigenvalue problem with an interface. The main aim of this paper is to prove the stability and convergence of an immersed finite element method (IFEM) for eigenvalues using Crouzeix–Raviart P1-nonconforming approximation. We show that spectral analysis for the classical eigenvalue problem can be easily applied to our model problem. We analyze the IFEM for elliptic eigenvalue problems with an interface and derive the optimal convergence of eigenvalues. Numerical experiments demonstrate our theoretical results.

Superconvergent Two-Grid Methods for Elliptic Eigenvalue Problems

Some numerical algorithms for elliptic eigenvalue problems are proposed, analyzed, and numerically tested. The methods combine advantages of the two-grid algorithm, two-space method, the shifted inverse power method, and the polynomial preserving recovery technique . Our new algorithms compare favorably with some existing methods and enjoy superconvergence property.

The lower bound property of the Morley element eigenvalues

In this paper, we prove that the Morley element eigenvalues approximate the exact ones from below on regular meshes, including adaptive local refined meshes, for the fourth-order elliptic eigenvalue problems with the clamped boundary condition in any dimension. And we implement the adaptive computation with the Morley element to obtain lower bounds of eigenvalues for the vibration problem of clamped plate under tension and a fourth-order elliptic eigenvalue problem with variable coefficients.

A combinational efficient algorithm for elliptic eigenvalue problems

In this paper, we present a combinational efficient algorithm for solving second-order elliptic ordinary differential equation eigenvalue problems on four kinds of meshes. Firstly, a combinational scheme is proposed by leveraging the linear finite element scheme and the second-order finite difference scheme based on mass lumping method. An efficient algorithm is constructed by using the combinational scheme with a constant combined parameter. Then, we improve the efficient algorithm by computing the quasi-optimal combined parameters for different eigenvalues. Finally, numerical experiments tested on both uniform and nonuniform meshes are shown to illustrate the efficiency of our algorithms.

Dynamics of a reaction–diffusion waterborne pathogen model with direct and indirect transmission

In this paper a reaction–diffusion waterborne pathogen model, which incorporates both direct and indirect transmission pathways, is proposed. Then the basic reproduction number R0 defined as the spectral radius of the next generation operator is established for the model system, and an equivalent characterization of R0 is obtained in terms of an elliptic eigenvalue problem. Furthermore, we show that R0 serves as a threshold parameter which predicts whether the disease will persist. By using numerical simulations, we study the influence of diffusion coefficients, spatial heterogeneity and multiple transmission pathways.

On the Existence of Infinite Sequences of Ordered Positive Solutions of Nonlinear Elliptic Eigenvalue Problems

Abstract In this work, we employ minimization arguments and topological degree theory for mappings of type S + ${S_{+}}$ to prove the existence of infinite sequences of ordered solutions for a class of nonlinear equations in which the semilinear term vanishes an infinite number of times. Our results extend earlier ones by Hess [10] and by Loc and Schmitt [15].

A minimization problem for an elliptic eigenvalue problem with nonlinear dependence on the eigenparameter

In this paper we examine an eigenvalue optimization problem. Given two nonlinear functions α(λ) and β(λ), find a subset D of the unit ball of measure A for which the first Dirichlet eigenvalue of the operator −div((α(λ)χD+β(λ)χDc)∇u)=λu is as small as possible. This sort of nonlinear eigenvalue problems arises in the study of some quantum dots taking into account an electron effective mass. We establish the existence of a solution, and we propose a numerical algorithm to obtain an approximate description of the optimizer.

A Homotopy Method for Finding All Solutions of a Multiparameter Eigenvalue Problem

Multiparameter eigenvalue problems arise naturally in a variety of applications, especially in certain boundary value problems when the technique of separation of variables is applied. The homotopy method has been successfully used to solve the classical eigenvalue problems, the generalized eigenvalue problems, the right definite two-parameter eigenvalue problems and the weakly elliptic two-parameter eigenvalue problems. In this paper, we propose applying homotopy methods to solve the general multiparameter eigenvalue problems. Comparing with the existing homotopy methods for multiparameter eigenvalue problems, a distinct feature of our proposed method is that it is suitable for multiparameter eigenvalue problems without any limitations on the special properties of the coefficient matrices. Comparing with the method of transforming the multiparameter eigenvalue problem into the simultaneous eigenvalue problem, our proposed method is more efficient for coefficient matrices of large order, and has some adva...

A Posteriori Error Estimates of a Weakly Over-Penalized Symmetric Interior Penalty Method for Elliptic Eigenvalue Problems

AbstractA weakly over-penalized symmetric interior penalty method is applied to solve elliptic eigenvalue problems. We derive a posteriori error estimator of residual type, which proves to be both reliable and efficient in the energy norm. Some numerical tests are provided to confirm our theoretical analysis.

Second Domain Variation for Problems with Robin Boundary Conditions

In this paper, the first and second domain variations for functionals related to elliptic boundary and eigenvalue problems with Robin boundary conditions are computed. Extremal properties of the ball among nearly spherical domains of given volume are derived. The discussion leads to a Steklov eigenvalue problem. As a by-product, a general characterization of the optimal shapes is obtained.

An analysis on lower bound and upper bound and construction of high accuracy schemes in PDE eigen-computation

Any conforming finite element scheme always gives an upper bound for elliptic eigenvalue problem. We point out that it is essentially equal to ( u h , u h ) ≦ ( u , u h )≦ ( u , u ), and a lower bound scheme wich is called a dual scheme with the finite element projection is proposed. If a ( u - u h , u h ) = O (λ|| u - u h || L 2 2 ), a series of high order schemes with the same accuracy have been obtained, such as harmonic means λ H h := (2 a ( u h , u h ))/(( u , u )+( u h , u h )). For the linear and multi-linear element of rectangle, triangle and parallelepiped domains, two key conditions are investigated: a ( u - u h , u h ) = O (|| u - u h || L 2 2 ) and the distance between the L 2 norm ( u , u ) and l 2 norm in the discrete grid of the eigen-functions. The numerical experiments are carried out to validate the theoretical results. For Laplace problem in 2D/3D we give some high order schemes (6th, 8th and up to 10th order) to compute the first dozens of eigen-values. They are efficient to the singular eigen-functions and high frequency eigen-values too.