Elliptic Eigenvalue Problems Research Articles

Analysis of a vector-borne disease model with vector-bias mechanism in advective heterogeneous environment

Abstract This study proposed and analyzed a vector-borne reaction–diffusion–advection model with vector-bias mechanism and heterogeneous parameters in one-dimensional habitat. The basic reproduction number R 0 {{\mathfrak{R}}}_{0} in connection with principal eigenvalue of elliptic eigenvalue problem is characterized as the role of determining the threshold dynamics of the system. The main objective of this study is to investigate the asymptotic profiles and monotonicity of R 0 {{\mathfrak{R}}}_{0} with respect to diffusion rates and advection rates under certain conditions. Through exploring the level set of R 0 {{\mathfrak{R}}}_{0} , we also find that there exists a unique surface separating the dynamics. Our results also reveal that the infected hosts and vectors will aggregate at the downstream end if the ratio of advection rates and diffusion rates is sufficiently large.

Analytic and Gevrey Class Regularity for Parametric Elliptic Eigenvalue Problems and Applications

Analytic and Gevrey Class Regularity for Parametric Elliptic Eigenvalue Problems and Applications

Nonlinear elliptic eigenvalue problems in cylindrical domains becoming unbounded in one direction

The aim of this work is to characterize the asymptotic behaviour of the first eigenfunction of the generalised p-Laplace operator with mixed (Dirichlet and Neumann) boundary conditions in cylindrical domains when the length of the cylindrical domains tends to infinity. This generalises an earlier work of Chipot et al. (Asymptot. Anal. 85(3–4) (2013) 199–227) where the linear case p = 2 is studied. Asymptotic behavior of all the higher eigenvalues of the linear case and the second eigenvalues of general case (using topological degree) for such problems is also studied.

Asymptotic profiles of a spatial vector‐borne disease model with Fokker–Planck‐type diffusion

AbstractThis paper is concerned with a spatially heterogeneous vector‐borne disease model that follows the Fokker–Planck‐type diffusion law. One of the significant features in our model is that Fokker–Planck‐type diffusion is used to characterize individual movement, which poses new challenges to theoretical analysis. We derive for the first time the variational characterization of basic reproduction ratio for the model under certain conditions and investigate its asymptotic profiles with respect to the diffusion rates. Furthermore, via overcoming the difficulty of the associated elliptic eigenvalue problem, the asymptotic behaviors of endemic equilibrium for the model are discussed. Our results imply that whether rapid or slow movement of susceptible and infected individuals are conducive to disease control depends on the degree of disease risk in the habitat. Numerically, we verify the theoretical results and detect that Fokker–Planck‐type diffusion may amplify the scale of disease infection, which in turn increases the complexity of disease transmission by comparing the impacts of distinct dispersal types on disease dynamics.

On the role of advection in a spatial epidemic model with general boundary conditions

This paper is concerned with a reaction-diffusion-advection model for vector-borne disease with general boundary conditions and general incidences. Due to the boundary conditions, we first apply the eigenvalue theory of elliptic system to prove the existence and uniqueness of steady state for the model. The well-posedness of the model is established using an induction argument. By overcoming the difficulty of the associated elliptic eigenvalue problem, we originally derive the variational expression of the basic reproduction ratio R0. The asymptotic profiles and monotonicity of R0 with respect to the mobility and advection rates are investigated following the variational characterization of R0. Furthermore, the spatial dynamics of the model with Robin type boundary condition are categorized via classifying the level set of R0. This work provides new clues for further research on the spread of epidemics in open advective environments.

Dynamics of a Reaction–Diffusion–Advection System with Nonlinear Boundary Conditions

In this paper, we consider a single-species reaction–diffusion–advection population model with nonlinear boundary condition in heterogenous space. We not only investigate the existence, nonexistence and stability of positive steady-state solutions through a linear elliptic eigenvalue problem by means of variational approach, but also verify the existence of steady-state bifurcations at zero solution through Crandall and Robinowitz bifurcation theory and discuss the stability of bifurcations, which can lead to Allee effect when the bifurcation is subcritical.

Dynamics on a degenerated reaction–diffusion Zika transmission model

In view of environmental transmission and spatial heterogeneity, a degenerated reaction–diffusion Zika transmission model is established and analyzed with more realistic factors. Besides the mosquito reproduction number Rm, a novel computation formula of the basic reproduction number of disease R0 in terms of the principal eigenvalue of an elliptic eigenvalue problem is provided. Dynamic analysis results show that the proposed system follows a threshold dynamics based on Rm or R0. Especially, for the critical case of R0=1, global attractiveness of the disease-free steady state is obtained by constructing a new Lyapunov function, which is usually one of challenges for some spatially heterogeneous models.

Modified-operator method for the calculation of band diagrams of crystalline materials

In solid state physics, electronic properties of crystalline materials are often inferred from the spectrum of periodic Schrödinger operators. As a consequence of Bloch’s theorem, the numerical computation of electronic quantities of interest involves computing derivatives or integrals over the Brillouin zone of so-called energy bands, which are piecewise smooth, Lipschitz continuous periodic functions obtained by solving a parametrized elliptic eigenvalue problem on a Hilbert space of periodic functions. Classical discretization strategies for resolving these eigenvalue problems produce approximate energy bands that are either non-periodic or discontinuous, both of which cause difficulty when computing numerical derivatives or employing numerical quadrature. In this article, we study an alternative discretization strategy based on an ad hoc operator modification approach. While specific instances of this approach have been proposed in the physics literature, we introduce here a systematic formulation of this operator modification approach. We derive a priori error estimates for the resulting energy bands and we show that these bands are periodic and can be made arbitrarily smooth (away from band crossings) by adjusting suitable parameters in the operator modification approach. Numerical experiments involving a toy model in 1D, graphene in 2D, and silicon in 3D validate our theoretical results and showcase the efficiency of the operator modification approach.

An Efficient Finite Element Method and Error Analysis Based on Dimension Reduction Scheme for the Fourth-Order Elliptic Eigenvalue Problems in a Circular Domain

In this paper, we propose an effective finite element method for the fourth order elliptic eigenvalue problems in a circular domain. First, by using polar coordinates transformation and the orthogonal property of Fourier basis functions, the original problem is turned into a series of equivalent one-dimensional eigenvalue problems. Second, according to the properties of Laplace operator in polar coordinate, we deduce the polar conditions and introduce suitable weighted Sobolev space. Based on the polar conditions and weighted Sobolev space, we establish the weak form and the corresponding discrete form. Third, we prove the error estimates of approximation eigenvalues and eigenvectors by means of the spectral theory of compact operators for each one-dimensional eigenvalue system. Finally, we provide some numerical experiments to show the validity of the algorithm and the correctness of the theoretical results.

A locally optimal preconditioned Newton-Schur method for symmetric elliptic eigenvalue problems

A locally optimal preconditioned Newton-Schur method is proposed for solving symmetric elliptic eigenvalue problems. Firstly, the Steklov-Poincaré operator is used to project the eigenvalue problem on the domain Ω \Omega onto the nonlinear eigenvalue subproblem on Γ \Gamma , which is the union of subdomain boundaries. Then, the direction of correction is obtained via applying a non-overlapping domain decomposition method on Γ \Gamma . Four different strategies are proposed to build the hierarchical subspace U k + 1 U_{k+1} over the boundaries, which are based on the combination of the coarse-subspace with the directions of correction. Finally, the approximation of eigenpair is updated by solving a local optimization problem on the subspace U k + 1 U_{k+1} . The convergence rate of the locally optimal preconditioned Newton-Schur method is proved to be γ = 1 − c 0 T h , H − 1 \gamma =1-c_{0}T_{h,H}^{-1} , where c 0 c_{0} is a constant independent of the fine mesh size h h , the coarse mesh size H H and jumps of the coefficients; whereas T h , H T_{h,H} is the constant depending on stability of the decomposition. Numerical results confirm our theoretical analysis.

Multilevel quasi-Monte Carlo for random elliptic eigenvalue problems I: regularity and error analysis

Abstract Stochastic partial differential equation (PDE) eigenvalue problems are useful models for quantifying the uncertainty in several applications from the physical sciences and engineering, e.g., structural vibration analysis, the criticality of a nuclear reactor or photonic crystal structures. In this paper we present a multilevel quasi-Monte Carlo (MLQMC) method for approximating the expectation of the minimal eigenvalue of an elliptic eigenvalue problem with coefficients that are given as a series expansion of countably-many stochastic parameters. The MLQMC algorithm is based on a hierarchy of discretizations of the spatial domain and truncations of the dimension of the stochastic parameter domain. To approximate the expectations, randomly shifted lattice rules are employed. This paper is primarily dedicated to giving a rigorous analysis of the error of this algorithm. A key step in the error analysis requires bounds on the mixed derivatives of the eigenfunction with respect to both the stochastic and spatial variables simultaneously. Under stronger smoothness assumptions on the parametric dependence, our analysis also extends to multilevel higher-order quasi-Monte Carlo rules. An accompanying paper (Gilbert, A. D. & Scheichl, R. (2023) Multilevel quasi-Monte Carlo methods for random elliptic eigenvalue problems II: efficient algorithms and numerical results. IMA J. Numer. Anal.) focusses on practical extensions of the MLQMC algorithm to improve efficiency and presents numerical results.

Analyticity of parametric elliptic eigenvalue problems and applications to quasi-Monte Carlo methods

In the present paper, we study the analyticity of the leftmost eigenvalue of the linear elliptic partial differential operators with random coefficient and analyse the convergence rate of the quasi-Monte Carlo method for approximation of the expectation of this quantity. The random coefficient is assumed to be represented by an affine expansion a 0 ( x ) + ∑ j ∈ N y j a j ( x ) , where elements of the parameter vector y = ( y j ) j ∈ N ∈ U ∞ are independent and identically uniformly distributed on U := [ − 1 2 , 1 2 ] . Under the assumption ‖ ∑ j ∈ N ρ j | a j | ‖ L ∞ ( D ) < ∞ with some positive sequence ( ρ j ) j ∈ N ∈ ℓ p ( N ) for p ∈ ( 0 , 1 ] we show that for any y ∈ U ∞ , the elliptic partial differential operator has a countably infinite number of eigenvalues ( λ j ( y ) ) j ∈ N which can be ordered non-decreasingly. Moreover, the spectral gap λ 2 ( y ) − λ 1 ( y ) is uniformly positive in U ∞ . From this, we prove the holomorphic extension property of λ 1 ( y ) to a complex domain in C ∞ and estimate partial derivatives of λ 1 ( y ) with respect to the parameter y by using Cauchy's formula for analytic functions. Based on these bounds we prove the dimension-independent convergence rate of the quasi-Monte Carlo method to approximate the expectation of λ 1 ( y ) .

Convergence Analysis of Newton–Schur Method for Symmetric Elliptic Eigenvalue Problem

In this paper, we consider the Newton–Schur method in Hilbert space and obtain quadratic convergence. For the symmetric elliptic eigenvalue problem discretized by the standard finite element method and nonoverlapping domain decomposition method, we use the Steklov–Poincaré operator to reduce the eigenvalue problem on the domain into the nonlinear eigenvalue subproblem on , which is the union of subdomain boundaries. We prove that the convergence rate for the Newton–Schur method is , where the constant is independent of the fine mesh size and coarse mesh size , and and are errors after and before one iteration step, respectively. For one specific inner product on , a sharper convergence rate is obtained, and we can prove that . Numerical experiments confirm our theoretical analysis.

A Nonnested Augmented Subspace Method for Elliptic Eigenvalue Problems with Curved Interfaces

A Nonnested Augmented Subspace Method for Elliptic Eigenvalue Problems with Curved Interfaces

Isoparametric numerical integration on enriched 4D simplicial elements

The enriched finite element method is a natural improvement of the classical finite element method for solving complicated engineering problems. It is an essential tool for successful treating difficulties, singularities, non-Lipschitz internal boundaries, curved boundaries with complex geometry, crack propagations, etc. This paper deals with an enrichment of the four-dimensional isoparametric quadratic Lagrangian finite element. The paper discusses a new approach in which bubble functions are replaced with specific cubic ones in the enrichment process. The new method is applied to simplicial elements in four-dimensional Euclidean space, where lumping of the mass matrix is obtained. Error estimates in numerical integration and the interpolation by the new elements are proved. Applications of the quadrature rules for solving elliptic eigenvalue problems are demonstrated. A new operator that maps tesseract onto a four-dimensional ball is defined. The new quadrature rules are tested by integrating Bessel functions and by solving a second-order elliptic eigenvalue problem with Dirichlet boundary conditions.

Analysis of a weak Galerkin method for second-order elliptic equations with minimal regularity on polytopal meshes

We propose an error analysis of a weak Galerkin method for second-order elliptic equations with minimal regularity requirements, i.e., the exact solution u belongs to H1+s(Ω) with s>0 possibly close to zero. In this case, the conormal derivative of u is not well-defined on the mesh faces, which results in a legitimacy problem of the error equation originally derived under smooth assumption. We employ the weak meaning of the conormal derivative introduced in [Ern, A., & Guermond, J. L. (2021). Quasi-optimal Nonconforming Approximation of Elliptic PDEs with Contrasted Coefficients andH1+s,s>0, Regularity. Foundations of Computational Mathematics, 1–36.] to establish new error equation and optimal a priori error estimates for low regular solutions on polytopal meshes. We also investigate the corresponding elliptic eigenvalue problems. The numerical experiments verify the theoretical results and agree with the benchmark eigenvalue problem.

Weighted local Weyl laws for elliptic operators

Let $A$ be an elliptic pseudo-differential operator of order $m$ on a closed manifold $\man$ of dimension $n>0$, formally positive self-ajdoint with respect to some positive smooth density $d\mu_\man$. Then, the spectrum of $A$ is made up of a sequence of eigenvalues $(\lambda_k)_{k\geq 1}$ whose corresponding eigenfunctions $(e_k)_{k\geq 1}$ are $C^\infty$ smooth. Fix $s\in\R$ and define \ \[ K_L^s(x,y)=\sum_{0<\lambda_k\leq L}\lambda_k^{-s} e_k(x)\overline{e_k(y)}\, . \] We derive asymptotic formulae near the diagonal for the kernels $K_L^s(x,y)$ when $L\rightarrow +\infty$ with fixed $s$. For $s=0$, $K^0_L$ is the kernel of the spectral projector studied by H\ormander in [Hor68]. In the present work we build on H\ormander's result to study the kernels $K^s_L$. If $s<\frac{n}{m}$, $K_L^s$ is of order $L^{-s+n/m}$ and near the diagonal, the rescaled leading term behaves like the Fourier transform of an explicit function of the symbol of $A$. If $s=\frac{n}{m}$, under some explicit generic condition on the principal symbol of $A$, which holds if $A$ is a differential operator, the kernel has order $\ln(L)$ and the leading term has a logarithmic divergence smoothed at scale $L^{-1/m}$. Our results also hold for elliptic differential Dirichlet eigenvalue problems.

Exponential ReLU Neural Network Approximation Rates for Point and Edge Singularities

In certain polytopal domains varOmega , in space dimension d=2,3, we prove exponential expressivity with stable ReLU Neural Networks (ReLU NNs) in H^1(varOmega ) for weighted analytic function classes. These classes comprise in particular solution sets of source and eigenvalue problems for elliptic PDEs with analytic data. Functions in these classes are locally analytic on open subdomains Dsubset varOmega , but may exhibit isolated point singularities in the interior of varOmega or corner and edge singularities at the boundary partial varOmega . The exponential approximation rates are shown to hold in space dimension d = 2 on Lipschitz polygons with straight sides, and in space dimension d=3 on Fichera-type polyhedral domains with plane faces. The constructive proofs indicate that NN depth and size increase poly-logarithmically with respect to the target NN approximation accuracy varepsilon >0 in H^1(varOmega ). The results cover solution sets of linear, second-order elliptic PDEs with analytic data and certain nonlinear elliptic eigenvalue problems with analytic nonlinearities and singular, weighted analytic potentials as arise in electron structure models. Here, the functions correspond to electron densities that exhibit isolated point singularities at the nuclei.

Dynamics of an HIV infection model with virus diffusion and latently infected cell activation

Effective combination therapy usually reduces the plasma viral load of HIV to below the detection limit, but it cannot eradicate the virus. The latently infected cell activation is considered to be the main obstacle to completely eradicating HIV infection. In this paper, we consider an HIV infection model with latently infected cell activation, virus diffusion and spatial heterogeneity under Neumann boundary condition. The basic reproduction ratio is characterized by the principal eigenvalue of the related elliptic eigenvalue problem. Besides, by constructing Lyapunov functionals and using Green’s first identity, the global threshold dynamics of the system are completely established. Numerical simulations are carried out to illustrate the theoretical results, in particular, the influence of virus diffusion rate on the basic reproduction ratio is addressed.

Spectral Element Methods for Eigenvalue Problems Based on Domain Decomposition

The spectral element method performs very well in solving PDE eigenvalue problems for some cases. In this paper, we propose and analyze parallel spectral element methods to solve elliptic PDE eigenvalue problems (although we use the Laplacian operator as a model) based on domain decomposition methods. The method demonstrates great advantage in both high accuracy and high efficiency by combining the spectral element method and parallel computing. In addition, the method could be performed in parallel with good scalability. We present a systematic construction and analysis for both the non-shifted scheme and the shifted scheme in 1D, 2D, and 3D. Furthermore, we obtain an optimal convergence rate of the eigenpair in the sense that it is independent of $h$ (the size of the spectral elements), $p_i$ (the degree of the polynomial in each interior spectral element $T_i$ of subdomains), and $N$ (the number of subdomains) by choosing a suitable overlapping size $\delta$. Numerical tests are provided to show the efficiency of the method.