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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 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).

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.

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.