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This article is devoted to the construction and study of the finite element method for solving a two-dimensional nonlinear equation of elliptic type. Equations of this type arise in solving many applied problems, including problems of the theory of multiphase filtering, the theory of semiconductor devices, and many others. The relevance of the study of this problem is associated with the need to develop effective parallel methods for solving this problem. To discretize the equation, a mixed finite element method with Brezzi-Douglas-Marini elements is used. The issue of the convergence of the finite element method is investigated. To linearize the equation, the Picard iterative method is constructed. Two classes of basis functions of finite elements are used in the paper. A comparative analysis of the effectiveness of several direct and iterative methods for solving the resulting system of linear algebraic equations is carried out, including the method based on the Bunch-Kaufman LDLt factorization, the method of minimal residuals, the symmetric LQ method, the stabilized biconjugate gradient method, and a number of other iterative Krylov subspace algorithms with preconditioners based on incomplete LU decomposition. The method has been tested on several model problems by comparing an approximate solution with a known exact solution. The results of the analysis of the method error in various norms depending on the diameter of the mesh are presented.

Recently, a nonlinear Poisson equation has been introduced to model nonlinear and nonlocal hyperpolarization effects in electrostatic solute-solvent interaction for biomolecular solvation analysis. Due to a strong nonlinearity associated with the heterogeneous dielectric media, this Poisson model is difficult to solve numerically, particularly for large protein systems. A new pseudo-transient continuation approach is proposed in this paper to efficiently and stably solve the nonlinear Poisson equation. A Douglas type alternating direction implicit (ADI) method is developed for solving the pseudo-time dependent Poisson equation. Different approximations to the dielectric profile in heterogeneous media are considered in the standard finite difference discretization. The proposed ADI scheme is validated by considering benchmark examples with exact solutions and by solvation analysis of real biomolecules with various sizes. Numerical results are in good agreement with the theoretical prediction, experimental measurements, and those obtained from the boundary value problem approach. Since the time stability of the proposed ADI scheme can be maintained even using very large time increments, it is efficient for electrostatic analysis involving hyperpolarization effects.

In this note, we study a Modica type gradient estimate for smooth solutions to general non-linear Poisson equation Δ u − f ( u ) = 0 , in M n , u : M n → R where ( M , g ) is a complete Riemannian manifold with bounded geometry and non-negative Ricci curvature and f is the derivative of the non-negative smooth function F ( u ) on R. Then we use this gradient estimate to conclude a Liouville theorem.

This paper introduces a novel method for deducing high-order compact difference schemes for the two-dimensional (2D) Poisson equation. Like finite volume method, a dual partition is introduced. Combining Simpson integral formula and parabolic interpolation, a family of fourth-order and sixth-order compact difference schemes are obtained based on three different types of dual partitions. Moreover, several new fourth-order compact schemes are gained and numerical experiments are shown two of them are much better than almost any other fourth-order schemes which have been presented in others’ work. The outline for the nonlinear Poisson equation is also given. Numerical experiments are presented to verify the feasibility of this new method and the high accuracy of these fourth-order and sixth-order compact difference schemes.

The numerical solution of the bi-dimensional nonlinear Poisson equations under Cauchy boundary conditions is considered. Using Green identity we show that this problem is equivalent to solve a bi-dimensional Fredholm integral equation of the first kind which can in turn be handled as a bi-dimensional generalized inverse moment problem. In the particular linear case the Helmholtz PDE is recovered and, within our scheme, the problem reduces to a bi-dimensional Hausdorff moment problem. In all the cases we find approximated solutions for the associated finite moment problems and bounds for the corresponding errors. Fil: Pintarelli, María Beatriz. Universidad Nacional de La Plata. Facultad de Ciencias Exactas. Departamento de Matemáticas; Argentina Fil: Vericat, Fernando. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - La Plata. Instituto de Física de Líquidos y Sistemas Biológicos. Universidad Nacional de La Plata. Facultad de Ciencias Exactas. Instituto de Física de Líquidos y Sistemas Biológicos; Argentina. Universidad Nacional de La Plata. Facultad de Ingeniería; Argentina

In this note we discuss how several results characterizing the qualitative behavior of solutions to the nonlinear Poisson equation can be generalized to harmonic maps with potential between complete Riemannian manifolds. This includes gradient estimates, monotonicity formulas, and Liouville theorems under curvature and energy assumptions.

Based on Newtonian dynamics, observations show that the luminous masses of astrophysical objects that are the size of a galaxy or larger are not enough to generate the measured motions which they supposedly determine. This is typically attributed to the existence of dark matter, which possesses mass but does not radiate (or absorb radiation). Alternatively, the mismatch can be explained if the underlying dynamics is not Newtonian. Within this conceptual scheme, Modified Newtonian Dynamics (MOND) is a successful theoretical paradigm. MOND is usually expressed in terms of a nonlinear Poisson equation, which is difficult to analyse for arbitrary matter distributions. We study the MONDian gravitational field generated by slightly non-spherically symmetric mass distributions based on the fact that both Newtonian and MONDian fields are conservative (which we refer to as the compatibility condition). As the non-relativistic version of MOND has two different formulations (AQUAL and QuMOND) and the compatibility condition can be expressed in two ways, there are four approaches to the problem in total. The method involves solving a suitably defined linear deformation potential, which generally depends on the choice of MOND interpolation function. However, for some specific form of the deformation potential, the solution is independent of the interpolation function., 30 pages, 2 figures

After a quick review of the Lane-Emden equation and its properties, a composite of two different polytropes is introduced and some of the consequences are explored. The results are used to build a nonlinear electromagnetism with non-singular, solitonic solutions as charged particles.

In this article, we study the properties of the hyperinterpolation operator on the unit disc D in ℝ 2 . We show how hyperinterpolation can be used in connection with the Kumar―Sloan method to approximate the solution of a nonlinear Poisson equation on the unit disc (discrete Galerkin method). A bound for the norm of the hyperinterpolation operator in the space C(D) is derived. Our results prove the convergence of the discrete Galerkin method in the maximum norm if the solution of the Poisson equation is in the class C 1,δ (D), δ > 0. Finally, we present numerical examples which show that the discrete Galerkin method converges faster than O(n ―k ), for every k ∈ N if the solution of the nonlinear Poisson equation is in C°°(D).

A nonlinear Poisson–Boltzmann equation with inhomogeneous Robin type boundary conditions at the interface between two materials is investigated. The model describes the electrostatic potential generated by a vector of ion concentrations in a periodic multiphase medium with dilute solid particles. The key issue stems from interfacial jumps, which necessitate discontinuous solutions to the problem. Based on variational techniques, we derive the homogenisation of the discontinuous problem and establish a rigorous residual error estimate up to the first-order correction.

We investigate the structure of self-gravitating polytropic stellar systems. We present a method which allows to obtain approximate analytical solutions, $\psi_{n+\epsilon}({\bf x})$, of the nonlinear Poisson equation with the polytropic index $n+\epsilon$, given the solution $\psi_n({\bf x})$ with the polytropic index n, for any positive or negative $\epsilon$ such that $|\epsilon|\ll1$. Application of this method to the spherically symmetric stellar polytropes with $n\simeq5$ yields the solutions which describe spatially bound systems if n5. A heuristic approximate expressions for the radial profiles are also presented. Due to the duality between stellar and gas polytropes, our results are valid for gaseous, self-gravitating polytropic systems (e.g., molecular clouds) with the index $\gamma\simeq 6/5$. Stability of such systems and observational consequences for both stellar and gaseous systems are discussed., Comment: LaTeX, emulateapj, 5 pages, 2 PS figures. ApJ,accepted version. Other works are at http://www.cita.utoronto.ca/~medvedev/

We study the zero-mass-electrons limits in the hydrodynamic models for plasmas consisting of electrons and ions. We prove, under suitable assumptions, that the evolution of the ions is governed by the Euler equations coupled with a nonlinear Poisson equation, as the mass of electrons tends to zero.

This work deals with a model for a mixture of charged constituents introduced in [W. Dreyer et al. Overcoming the shortcomings of the Nernst-Planck model. emphPhys. Chem. Chem. Phys., 15:7075-7086, 2013]. The aim of this paper is to give a first existence and uniqueness result for the equilibrium situation. A main difference to earlier works is a momentum balance involving the gradient of pressure and the Lorenz force which persists in the stationary situation and gives rise to the dependence of the chemical potentials on the particle densities of every species.

We consider the steady-state two-dimensional motion of an inviscid incompressible fluid which obeys a nonlinear Poisson equation. By seeking solutions of a specific form, we arrive at some interesting new nonlinear vortex solutions.

The present paper deals with a generalization of the alternating-direction implicit (ADI) method for the two-dimensional nonlinear Poisson equation in a rectangular domain with integral boundary condition in one coordinate direction. The analysis of results of computational experiments is presented.

This paper gives a concise overview of some existing methods for the solution of the nonlinear Poisson equation in semiconductors. A method for the solution of this equation was recently proposed in this journal by I. D. Mayergoyz [J. Appl. Phys. 59, 195 (1986)]. Soon afterwards, an improved version was described by W. Keller [J. Appl. Phys. 61, 5189 (1987)]. Both methods are classified within the perspective of the existing methods. Moreover, Keller’s method is further improved by the introduction of scaled variables and by using red‐black ordening to allow for overrelaxation. All advantages of the two methods are maintained. An illustrative example shows an improvement in solution speed of at least a factor of 5.6.

A large anisotropy in the electrical conductivity of CdS appearing below 30\ifmmode^\circ\else\textdegree\fi{}K is reported. This anisotropy casts serious doubt on the customary view that impurity hopping supplies the low-temperature conductivity. We present a new model for this conductivity based upon the electronic screening of the piezoelectric polarization around dislocations. Numerical solutions of the highly nonlinear Poisson equation quantitatively justify this model.

Allowance for traps in certain space‐charge environments produces changes in the mathematical formalism which make it difficult to determine spatial variations in potential and the negative‐positive charge distribution. The trap‐free model for the blocking electrode can be extended into this realm by appropriate accommodation of the equations of continuity for specified kinetics of the trap system. The tendency of traps seems to be a moderation of the spatial fall‐off of potential with more gradual build‐up and decay of the charge densities. The nonlinear Poisson equation has been solved analytically by the development of powerful series methods, demonstrating that machine integration can be avoided.

An existence theorem is proved for a two-dimensional free boundary problem for a nonlinear Poisson equation. The method is constructive, employing a monotone iterative scheme, and it provides classical solutions.