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In this paper, a coupled phase-field (PF) and lattice Boltzmann method (LBM) is presented to model the multiphysics phenomenon involving electro-chemical deposition. The deposition (or dissolution) of the electrode is represented using variations of an order-parameter. The time-evolution of an order-parameter is proportional to the variation of a Ginzburg–Landau free-energy functional. Further, the free-energy densities of the two phases are defined based on a dilute or an ideal solution approximation. An efficient LBM is used to obtain the converged electro-static potential field for each physical time-step of the evolution of the PF variable. The coupled approach demonstrates the applicability of the LBM in a multiphysics scenario. The numerical validation for the coupled approach is performed by the simulation of the electrodeposition process of Cu from CuSO₄ solution.

We present a new approach to find accurate solutions to the Poisson equation, as obtained from the steady-state limit of a diffusion equation with strong source terms. For this purpose, we start from Boltzmann's kinetic theory and investigate the influence of higher-order terms on the resulting macroscopic equations. By performing an appropriate expansion of the equilibrium distribution, we provide a method to remove the unnecessary terms up to a desired order and show that it is possible to find, with high level of accuracy, the steady-state solution of the diffusion equation for sizeable Knudsen numbers. In order to test our kinetic approach, we discretize the Boltzmann equation and solve the Poisson equation, spending up to six order of magnitude less computational time for a given precision than standard lattice Boltzmann methods (LBMs).

A solution of the Coulomb problem in Fourier representation is given. The problems with the long-range parts of the Coulomb potential are circumvented by a separate handling of them. This allows the Fourier representation to develop its typical efficiency and robusteness.

The lattice-Bhatnagar-Gross-Krook (BGK) method has been used to simulate fluid flow in the nearly incompressible limit. But for the completely incompressible flows, two special approaches should be applied to the general model, for the steady and unsteady cases, respectively. Introduced by Zou et al.,¹ the method for steady incompressible flows will be described briefly in this paper. For the unsteady case, we will show, using a simple numerical example, the need to solve a Poisson equation for pressure.

The analytical formula for calculating the Coulomb energy of spherical nucleus with Woods–Saxon charge distribution is refined by taking into account the higher-order terms of surface diffuseness. The obtained relative deviations are smaller than 0.05% from the results with numerical integration for almost all calculated nuclei.

We previously proposed a formula for calculating the Coulomb energy of spherical nucleus with Woods–Saxon charge distribution. In this work, the analytical formula is extended for description of the Coulomb energy of nucleus with β₂ and β₄ deformation.

In this paper, we have investigated the Thomas–Fermi model for the electron gas in Rindler space. We have observed that if the uniform acceleration is along $+x$-direction, then there is $y-z$-symmetry in space. For the sake of mathematical simplicity, we have therefore assumed a two-dimensional spatial structure $(x-y)$ in Rindler space. It has been shown that in two-dimensional spatial coordinates, the electrons are distributed discontinuously but in a periodic manner in a number of rectangular strip-like domains along $\pm y$-direction. Some of them are having void structure, with no electrons inside such rectangular strips, while some are filled up with electrons. We call the latter type domain as the normal zone. We have also given physical interpretation for such exotic type electron distribution in Rindler space.

As a widely used equation in electrostatics, the Poisson equation has significant research value in numerical solution. The basic principle of existing methods is to divide the solution domain into various grids and solve the numerical solutions at each grid node. Therefore, the accuracy of the solution is strongly correlated with the grid density divided. Based on this, this paper proposes a grid-free numerical calculation method that requires far fewer model parameters than traditional methods, and can ignore the order of the equation to solve high-dimensional Poisson equations. Given a Poisson equation, which has a certain type of boundary condition. A certain number of coordinate points are selected on the solution space and its boundary to construct a dataset. Using automatic differentiation technique to fit the differential operator in the equation, a loss function is constructed by incorporating the given boundary conditions or initial conditions, and the final numerical solution is obtained through iterative optimization algorithms. In the numerical experiment section, the algorithm proposed in this paper was used to solve the two-dimensional and three-dimensional Poisson equations with given exact solutions. The relative errors between the numerical solution and the true solution were $8.53e-4$ and $6.40e-2$, which are within the acceptable range. This proves that the proposed algorithm is feasible for solving the two-dimensional and three-dimensional Poisson equations with precise solutions. Secondly, the proposed algorithm is used to solve the four-dimensional Poisson equation with first-type boundary conditions, and the relative error range of the solution was within [0,0.56], which successfully extends the algorithm to solve high-dimensional Poisson equations and verifies its feasibility and efficiency in solving high-dimensional Poisson equations regardless of the dimension restriction.

The paper is devoted to the analysis of a drift-diffusion-Schrödinger–Poisson (DDSP) system. From the physical point of view, it describes the transport of a quasi-bidimensional electron gas confined in a nanostructure. Existence, uniqueness and long-time behavior of a weak solution were already obtained in Ref. 8 for constant scalar diffusion matrices. Here, we develop an L log L existence theory for the DDSP system for a general class of smooth diffusion matrices. Our argument relies on a Trudinger estimate for the entropy functional and a sharp bound on the Hamiltonian's spectrum.

We consider the problem of reinforcing an elastic medium by a strong, rough, thin external layer. This model is governed by the Poisson equation with homogeneous Dirichlet boundary condition. We characterize the asymptotic behavior of the solution as the shear modulus of the layer goes to infinity. We find that there are four types of behaviors: the limiting solution satisfies Poisson equation with Dirichlet boundary condition, Robin boundary condition or Neumann boundary condition, or the limiting solution does not exist. The specific type depends on the integral of the load on the medium, the curvature of the interface and the scaling relations among the shear modulus, the thickness and the oscillation period of the layer.

The main objective of this paper is to investigate an extension of the "Volterra-Gross" Laplacian on nuclear algebra of generalized functions. In so doing, without using the renormalization procedure, this extension provides a continuous nuclear realization of the square white noise Lie algebra obtained by Accardi–Franz–Skeide in Ref. 2. An extended-Gross diffusion process driven by a class of Itô stochastic equations is studied, and solution of the related Poisson equations is derived in terms of a suitable λ-potential.

This paper presents a new methodology for the deformation of soft objects by drawing an analogy between Poisson equation and elastic deformation. The potential energy stored in an elastic body as a result of a deformation caused by an external force is propagated among mass points by Poisson equation. An improved Poisson model is developed for propagating the energy generated by the external force in a natural manner. A cellular neural network (CNN) model is established to solve the Poisson model for the real-time requirement of soft object deformation. A haptic virtual reality system has been established for deformation simulation with force feedback. This proposed methodology not only deals with local and large-range deformations, but also accommodates isotropic, anisotropic and inhomogeneous materials by simply modifying constitutive coefficients, as well as predicts the mechanical behaviors of living tissues.

We study the regularity of solutions of the Poisson equation with Dirichlet, Neumann and mixed boundary values in polyhedral cones $K\subset {ℝ}^3$ in the specific scale $B_{\tau ,\tau }^{\alpha },\frac{1}{\tau }=\frac{\alpha }{d}+\frac{1}{p}$ of Besov spaces. The regularity of the solution in these spaces determines theorder of approximation that can be achieved by adaptive and nonlinear numerical schemes. We aim for a thorough discussion of homogeneous and inhomogeneous boundary data in all settings studied and show that the solutions are much smoother in this specific Besov scale compared to the fractional Sobolev scale $H^s$ in all cases,which justifies the use of adaptive schemes.

This paper describes a wavelet-based preconditioning technique for conjugate gradient method for linear systems derived from the Poisson equation. The linear systems solved with a conventional iterative matrix solver resulted in a marked increase in computing time with respect to an increase in grid points. Use of our wavelet-based technique leads to a matrix with a bounded condition number so that computing time is reduced significantly. In this study, one of the simplest wavelets, the Haar wavelet, is used for the purpose of developing a simple but efficient preconditioning algorithm. Simple wavelets having low data communication property such as the Haar wavelet are expected to be suitable for the purpose of improving computing performance. In this study, we also pay attention to the basic characteristics of the Haar-wavelet-based preconditioning method for a Poisson equation solver.

In this paper, we develop a collocation method for solving three-dimensional partial differential equations using Haar wavelet and Kronecker tensor product. The approach is based on a series of Haar wavelet basis functions to approximate sixth-order mixed derivatives. The proposed method is mathematically fast, less error and straightforward for the numerical solution of many types of three-dimensional Poisson, biharmonic and Helmholtz equations. Some numerical examples verify the accuracy and efficiency of the proposed method. Finally, we conclude that numerical results computed by our proposed method are more accurate than numerical results obtained in the existing methods in the literature. We find that the CPU time consumed by the suggested approach is lesser than the CPU time of existing methods. Thus, the process is fast, efficient and has a low numerical error.

In this work, we propose a nonconforming domain decomposition resolution scheme for partial differential equations. In this scheme, the discrete Lagrange multiplier is forced to vanish into cross-points between artificial interfaces. The resulting saddle point problem is then solved efficiently by a recently developed alternating oblique projection method. The properties of the proposed discretization scheme will be illustrated solving Poisson equations.

Interpolation techniques based on the local radial basis function differential quadrature (LRBF-DQ) method are proposed to interpolate the field values with arbitrarily given scattered data. The interpolation of more unknown field data from the limited known data is becoming an important research subject nowadays for the large-scale engineering analysis and design problems. Three new methods that utilize the field gradient, the governing equation, and both of them are undertaken. The geometric characteristics of the field values can therefore be taken into consideration by the field gradient. In the meantime the physical principle is able to be satisfied with the introduction of the governing equation whose merit is absent in the traditional interpolation techniques, such as the linear polynomial fitting (LPF) and the quadratic polynomial fitting (QPF) methods. The last method will take the advantages of both the methods of gradient and governing equation, namely the geometric characteristics and physical laws. Two numerical examples governed respectively by the three-dimensional (3D) Poisson and the 3D advection–diffusion equations are performed to demonstrate the accuracy and the stability of the present methods. The results are compared with those by the LPF and the QPF methods which show the current interpolation methods are more accurate and robust than the conventional ones.

In the vortex-in-cell method combined with the penalization method, fluid particles are traced by continuously updating their positions and strengths from the grid solution. To evaluate particle velocity, the velocity field is computed by solving a Poisson equation for the vector potential, namely ${\nabla }^2\psi =-\omega$, where $u=\nabla \times \psi$, and its computation can be greatly accelerated by the use of a fast Poisson solver. While this method offers an efficient way to simulate unsteady viscous flows, the computation of the boundary values when solving the Poisson equation can become a computation time bottleneck. Although adopting the fast multipole method can lead to saving further computation time, its disadvantage is that it requires complicated hierarchical data structures such as a quad-tree and oct-tree. In this paper, we introduce and assess an approximation method for specifying domain boundary values using splines. Using the proposed spline approximation method we achieve significant savings in both computation time and memory consumption.

In this paper, a meshless method based on finite point set is presented for solving the biharmonic equation with simply supported boundary condition. The biharmonic equation is split into a coupled system of two Poisson equations by introducing an intermediate function. The system of two Poisson equations is then solved by finite pointset method. This method is a local iterative method based on the weighted least square approximation. The advantage of this method is that two resultant of sizes only $6\times 6$ matrices are solved at each particle for the original and intermediate solution. Numerical results indicate a good accuracy of the finite pointset method.

We find a characterization of a sphere using a compact gradient almost Ricci soliton and the lower bound on the integral of Ricci curvature in the direction of potential field. Also, we use Poisson equation on a compact gradient almost Ricci soliton to find a characterization of the unit sphere.