Standard Laplace Operator Research Articles

Energy dissipation and maximum bound principle preserving scheme for solving a nonlocal ternary Allen-Cahn model

We present a new energy dissipation and maximum bound principle preserving scheme for solving a nonlocal ternary Allen-Cahn (NtAC) model, where the standard Laplace operator is deliberately replaced with a spatial convolution term that aims at describing long-range interactions among particles and the structure-preserving scheme is obtained based on the operator splitting method. The discrete maximum bound principle and global convergence of the new scheme are analyzed rigorously. Moreover, the scheme is strictly dissipative for a modified energy, which coincides with the original energy up to O(τ), where τ is the time step. To the best of our knowledge, this is the first work to show that the operator splitting method can guarantee the maximum bound principle for a ternary Allen-Cahn model. Various 2D/3D numerical experiments including the NtAC model with mass conservation are tested to verify the bound-preserving and energy-stable properties of the resulted scheme.

Stability and error estimates of Strang splitting method for the nonlocal ternary conservative Allen–Cahn model

The nonlocal model has attracted a great attention in materials science for describing various types of material heterogeneities and defects. In this study, we consider a nonlocal ternary conservative Allen–Cahn model, where the standard Laplace operator is intentionally replaced with a spatial convolution term that aims at describing long-range interactions among particles. A linear energy stable scheme is developed based on the operator splitting method. The mass conservation, energy stability and global convergence of the new scheme are analyzed rigorously. Numerical stability and convergence of the present numerical scheme are analyzed theoretically. Two and three dimensional numerical experiments are performed to validate the theoretical analysis and the efficiency of the method.

Benchmark Problems for the Numerical Discretization of the Cahn–Hilliard Equation with a Source Term

In this paper, we present benchmark problems for the numerical discretization of the Cahn–Hilliard equation with a source term. If the source term includes an isotropic growth term, then initially circular and spherical shapes should grow with their original shapes. However, there is numerical anisotropic error and this error results in anisotropic evolutions. Therefore, it is essential to use isotropic space discretization in the simulation of growth phenomenon such as tumor growth. To test numerical discretization, we present two benchmark problems: one is the growth of a disk or a sphere and the other is the growth of a rotated ellipse or a rotated ellipsoid. The computational results show that the standard discrete Laplace operator has severe grid orientation dependence. However, the isotropic discrete Laplace operator generates good results.

Minimal Gap in the Spectrum of the Sierpiński Gasket

Abstract This paper studies the size of the minimal gap between any two consecutive eigenvalues in the Dirichlet and in the Neumann spectrum of the standard Laplace operator on the Sierpiński gasket. The main result shows the remarkable fact that this minimal gap is achieved and coincides with the spectral gap. The Dirichlet case is more challenging and requires some key observations in the behavior of the dynamical system that describes the spectrum.

Deflated preconditioned Conjugate Gradient methods for noise filtering of low-field MR images

We study efficient implicit methods to denoise low-field MR images using a nonlinear diffusion operator as a regularizer. This problem can be formulated as solving a nonlinear reaction–diffusion equation. After discretization, a lagged-diffusion approach is used which requires a linear system solve in every nonlinear iteration. The choice of diffusion model determines the denoising properties, but it also influences the conditioning of the linear systems. As a solution method, we use Conjugate Gradient (CG) in combination with a suitable preconditioner and deflation technique. We consider four different preconditioners in combination with subdomain deflation. We evaluate the methods for four commonly used denoising operators: standard Laplace operator, two Perona–Malik type operators, and the Total Variation (TV) operator. We show that a Discrete Cosine Transform (DCT) preconditioner works best for problems with a slowly varying diffusion coefficient, while Jacobi preconditioning with subdomain deflation works best for a strongly varying diffusion, as happens for the TV operator. This research is part of a larger effort that aims to provide low-cost MR imaging capabilities for low-resource settings. We have evaluated the algorithms on low-field MRI images using inexpensive commodity hardware. With a suitable preconditioner for the chosen diffusion model, we are able to limit the time to denoise three-dimensional images of more than 2 million pixels to less than 15 s, which is fast enough to be used in practice.

A phase-field model and its efficient numerical method for two-phase flows on arbitrarily curved surfaces in 3D space

Herein, we present a phase-field model and its efficient numerical method for incompressible single and binary fluid flows on arbitrarily curved surfaces in a three-dimensional (3D) space. An incompressible single fluid flow is governed by the Navier–Stokes (NS) equation and the binary fluid flow is governed by the two-phase Navier–Stokes–Cahn–Hilliard (NSCH) system. In the proposed method, we use a narrow band domain to embed the arbitrarily curved surface and extend the NSCH system and apply a pseudo-Neumann boundary condition that enforces constancy of the dependent variables along the normal direction of the points on the surface. Therefore, we can use the standard discrete Laplace operator instead of the discrete Laplace–Beltrami operator. Within the narrow band domain, the Chorin’s projection method is applied to solve the NS equation, and a convex splitting method is employed to solve the Cahn–Hilliard equation with an advection term. To keep the velocity field tangential to the surface, a velocity correction procedure is applied. An effective mass correction step is adopted to preserve the phase concentration. Computational results such as convergence test, Kevin–Helmholtz instability, and Rayleigh–Taylor instability on curved surfaces demonstrate the accuracy and efficiency of the proposed method.

Bi-spatial Pullback Attractors of Fractional Nonclassical Diffusion Equations on Unbounded Domains with (p, q)-Growth Nonlinearities

This article is concerned with the existence as well as regularity of pullback attractors for a wide class of non-autonomous, fractional, nonclassical diffusion equations with (p, q)-growth nonlinearities defined on unbounded domains. We first establish the well-posedness of solutions as well as the existence of an energy equation, and then prove the existence of a unique pullback attractor in the fractional Sobolev space $$H^s({\mathbb {R}}^N)$$ for all $$s\in (0,1]$$. Finally, we show that this attractor is a bi-spatial $$(H^s({\mathbb {R}}^N),L^r({\mathbb {R}}^N))$$-attractor in two cases: $$\begin{aligned} 2\le r\le \max (p,q)\text { for }s=1;\text { and }2\le r<\max (p,q) \text { for }s\in (0,1). \end{aligned}$$The idea of energy equations and the method of asymptotic a priori estimates are employed to establish the pullback asymptotic compactness of the solutions in $$H^s({\mathbb {R}}^N)\cap L^r({\mathbb {R}}^N)$$ in order to overcome the non-compactness of the Sobolev embedding on the unbounded domain as well as the weak dissipation of the equations. This is the first time to study the bi-spatial attractor of the equation when the initial space is $$H^s({\mathbb {R}}^N)$$, and the result of this article is new even in $$H^1({\mathbb {R}}^N)\cap L^r({\mathbb {R}}^N)$$ when the fractional Laplace operator reduces to the standard Laplace operator.

Asymptotic behavior of fractional nonclassical diffusion equations driven by nonlinear colored noise on

This paper is concerned with the asymptotic behavior of the solutions of the fractional nonclassical diffusion equations driven by nonlinear colored noise defined on the entire space . We first establish the existence of energy equations for the solutions in with , and then prove the existence and uniqueness of pullback random attractors in when the nonlinear drift and diffusion terms have polynomial growth of arbitrary order. In addition, for linear additive noise, we show the upper semi-continuity of these attractors as the correlation time of the colored noise approaches zero. The idea of energy equations due to Ball is employed to establish the pullback asymptotic compactness of the solutions in in order to overcome the weak dissipativeness of the equation as well as the non-compactness of Sobolev embeddings on unbounded domains. The result of this paper is new even in the space when the fractional Laplace operator reduces to the standard Laplace operator.

The standard Laplace operator

The standard Laplace operator is a generalization of the Hodge Laplace operator on differential forms to arbitrary geometric vector bundles, alternatively it can be seen as generalization of the Casimir operator acting on sections of homogeneous vector bundles over symmetric spaces to general Riemannian manifolds. Stressing the functorial aspects of the standard Laplace operator $$\Delta $$ with respect to the category of geometric vector bundles we show that the standard Laplace operator commutes not only with all homomorphisms, but also with a large class of natural first order differential operators between geometric vector bundles. Several examples are included to highlight the conclusions of this article.

Bochner–Riesz profile of anharmonic oscillator [formula omitted

We investigate spectral multipliers, Bochner–Riesz means and the convergence of eigenfunction expansion corresponding to the Schrödinger operator with anharmonic potential L=−d2dx2+|x|. We show that the Bochner–Riesz profile of the operator L completely coincides with such profile of the harmonic oscillator H=−d2dx2+x2. It is especially surprising because the Bochner–Riesz profile of the one-dimensional standard Laplace operator is known to be essentially different and the case of operators H and L resembles more the profile of multidimensional Laplace operators. Another surprising element of the main obtained result is the fact that the proof is not based on restriction type estimates and instead an entirely new perspective has to be developed to obtain the critical exponent for Bochner–Riesz means convergence.

Fractional schrödinger equation with zero and linear potentials

Abstract This paper is about the fractional Schrödinger equation expressed in terms of the Caputo time-fractional and quantum Riesz-Feller space fractional derivatives for particle moving in a potential field. The cases of free particle (zero potential) and a linear potential are considered. For free particle, the solution is obtained in terms of the Fox H-function. For the case of a linear potential, the separation of variables method allows the fractional Schrödinger equation to be split into space fractional and time fractional ones. By using the Fourier and Mellin transforms for the space equation and the contour integrals technique for the time equation, the solutions are obtained also in terms of the Fox H-function. Moreover, some recent results related to the time fractional equation have been revised and reconsidered. The results obtained in this paper contain as particular cases already known results for the fractional Schrödinger equation in terms of the quantum Riesz space-fractional derivative and standard Laplace operator.

Restriction estimates, sharp spectral multipliers and endpoint estimates for Bochner-Riesz means

We consider abstract non-negative self-adjoint operators on L 2(X) which satisfy the finite-speed propagation property for the corresponding wave equation. For such operators, we introduce a restriction type condition, which in the case of the standard Laplace operator is equivalent to (p, 2) restriction estimate of Stein and Tomas. Next, we show that in the considered abstract setting, our restriction type condition implies sharp spectral multipliers and endpoint estimates for the Bochner-Riesz summability. We also observe that this restriction estimate holds for operators satisfying dispersive or Strichartz estimates. We obtain new spectral multiplier results for several second order differential operators and recover some known results. Our examples include Schrodinger operators with inverse square potentials on R n , the harmonic oscillator, elliptic operators on compact manifolds, and Schr¨odinger operators on asymptotically conic manifolds.

A generalized Poisson solver for first-principles device simulations

Electronic structure calculations of atomistic systems based on density functional theory involve solving the Poisson equation. In this paper, we present a plane-wave based algorithm for solving the generalized Poisson equation subject to periodic or homogeneous Neumann conditions on the boundaries of the simulation cell and Dirichlet type conditions imposed at arbitrary subdomains. In this way, source, drain, and gate voltages can be imposed across atomistic models of electronic devices. Dirichlet conditions are enforced as constraints in a variational framework giving rise to a saddle point problem. The resulting system of equations is then solved using a stationary iterative method in which the generalized Poisson operator is preconditioned with the standard Laplace operator. The solver can make use of any sufficiently smooth function modelling the dielectric constant, including density dependent dielectric continuum models. For all the boundary conditions, consistent derivatives are available and molecular dynamics simulations can be performed. The convergence behaviour of the scheme is investigated and its capabilities are demonstrated.

HOMOGENIZATION OF THE SCHRÖDINGER EQUATION WITH LARGE, RANDOM POTENTIAL

We study the behavior of solutions to a Schrödinger equation with large, rapidly oscillating, mean zero, random potential with Gaussian distribution. We show that in high dimension d &gt; 𝔪, where 𝔪 is the order of the spatial pseudo-differential operator in the Schrödinger equation (with 𝔪 = 2 for the standard Laplace operator), the solution converges in the L2 sense uniformly in time over finite intervals to the solution of a deterministic Schrödinger equation as the correlation length ε tends to 0. This generalizes to long times the convergence results obtained for short times and for the heat equation in [2]. The result is based on the decomposition of multiple scattering contributions introduced in [6]. In dimension d &lt; 𝔪, the random solution converges to the solution of a stochastic partial differential equation; see [1, 13].

Quasi-isospectrality on Quantum Graphs

Consider two quantum graphs with the standard Laplace operator and non-Robin type boundary conditions at all vertices. We show that if their eigenvalue-spectra agree everywhere aside from a sufficiently sparse set, then the eigenvalue-spectra and the length-spectra of the two quantum graphs are completely identical. Similarly, if their length-spectra agree everywhere aside from a sufficiently sparse set, then the quantum graphs have the same eigenvalue-spectrum and length-spectrum.

Novel Laplacian scheme and multiresolution modal curvatures for structural damage identification

Modal curvature is more sensitive to structural damage than directly measured mode shape, and the standard Laplace operator is commonly used to acquire the modal curvatures from the mode shapes. However, the standard Laplace operator is very prone to noise, which often leads to the degraded modal curvatures incapable of identifying damage. To overcome this problem, a novel Laplacian scheme is proposed, from which an improved damage identification algorithm is developed. The proposed step-by-step procedures in the algorithm include: (1) By progressively upsampling the standard Laplace operator, a new Laplace operator is constructed, from which a Laplace operator array is formed; (2) by applying the Laplace operator array to the retrieved mode shape of a damaged structure, the multiresolution curvature mode shapes are produced, on which the damage trait, previously shadowed under the standard Laplace operator, can be revealed by a ridge of multiresolution modal curvatures; (3) a Gaussian filter is then incorporated into the new Laplace operator to produce a more versatile Laplace operator with properties of both the smoothness and differential capabilities, in which the damage feature is effectively strengthened; and (4) a smoothened nonlinear energy operator is introduced to further enhance the damage feature by eliminating the trend interference of the multiresolution modal curvatures, and it results in a significantly improved damage trait. The proposed algorithm is tested using the data generated by an analytical crack beam model, and its applicability is validated with an experimental program of a delaminated composite beam using scanning laser vibrometer (SLV) to acquire mode shapes. The results are compared in each step, showing increasing degree of improvement for damage effect. Numerical and experimental results demonstrate that the proposed novel Laplacian scheme provides a promising damage identification algorithm, which exhibits apparent advantages (e.g., high-noise insusceptibility, insightful in damage revealment, and visualized damage presentation) over the standard Laplace operator.

The Goursat problem for a generalized Helmholtz operator in the plane

We consider the Goursat problem in the plane for partial differential operators whose principal part is the pth power of the standard Laplace operator. The data is posed on a union of 2p distinct lines through the origin. We show that the solvability of this Goursat problem depends on Diophantine properties of the geometry of lines on which the data is posed.

On the spectrum of the negative Laplacian for general doubly‐connected bounded domains

This paper is devoted to asymptotic formulas for functions related with the spectrum of the standard Laplace operator in two and three dimensional bounded doubly connected domains with impedance boundary conditions, where the impedances are assumed to be positive functions. Moreover, asymptotic expressions for the difference of eigenvalues related to impedance boundary value problems with different impedances are derived. Further results may be obtained.