Poisson-type Problems Research Articles

Multiplicity of Solutions for a Class of Kirchhoff–Poisson Type Problem

In this paper, we use the fountain theorems to investigate a class of nonlinear Kirchhoff–Poisson type problem. When the nonlinearity f satisfies the Ambrosetti–Rabinowitz’s 4-superlinearity condition, or under some weaker superlinearity condition, we establish two theorems concerning with the existence of infinitely many solutions.

Anisotropic p-Laplace Equations on long cylindrical domain

The main aim of this article is to study the Poisson type problem for anisotropic \(p\)-Laplace type equation on long cylindrical domains. The rate of convergence is shown to be exponential, thereby improving earlier known results for similar type of operators. The Poincar� inequality for a pseudo \(p\)-Laplace operator on an infinite strip-like domain is also studied and the best constant, like in many other situations in literature for other operators, is shown to be the same with the best Poincar� constant of an analogous problem set on a lower dimension.

Solving Nonlinear Elliptic PDEs in 2D and 3D Using Polyharmonic Splines and Low-Degree of Polynomials

In this paper, the improved localized method of approximated particular solutions (ILMAPS) using polyharmonic splines (PHS) together with a low-degree of polynomial basis is used to approximate solutions of various nonlinear elliptic Partial Differential Equations (PDEs). The method is completely meshfree, and it uses a radial basis function (RBF) that has no shape parameters. The discretization process is done through a simple collocation technique on a set of points in the local domain of influence. Resulted system of nonlinear algebraic equations is solved by the Picard method.The performance of the proposed method is tested on various nonlinear elliptical problems, including the Poisson-type problems in 2D and 3D with constant or variable coefficients on rectangular or irregular domains and the Poisson–Boltzmann equation with Dirichlet boundary conditions or mixed boundary conditions. The effect of domain shapes in 2D and 3D, types of boundary conditions, and degrees of PHS, and order of polynomial basis are examined. The performance of the method is compared with other bases such as multiquadrics (MQ) basis functions, and with results reported in the literature (method of particular solutions using polynomials). The numerical experiments suggest that ILMAPS with polyharmonic splines yields considerably superior accuracy than other RBFs as well as other approaches reported in the literature for solving nonlinear elliptic PDEs.

A Weighted Setting for the Numerical Approximation of the Poisson Problem with Singular Sources

We consider the approximation of Poisson type problems where the source is given by a singular measure and the domain is a convex polygonal or polyhedral domain. First, we prove the well-posedness ...

An Efficient and Accurate Numerical Method for the Spectral Fractional Laplacian Equation

We propose in this paper an efficient and accurate numerical method for the spectral fractional Laplacian equation using the Caffarelli–Silvestre extension. In particular, we propose several strategies to deal with the singularity and the additional dimension associated with the extension problem: (i) reducing the $$d+1$$ dimensional problem to a sequence of d-dimension Poisson-type problems by using the matrix diagonalizational method; (ii) resolving the singularity by applying the enriched spectral method in the extended dimension. We carry out rigorous analysis for the proposed numerical method, and provide abundant numerical examples to verify the theoretical results and illustrate effectiveness of the proposed method.

Dual reciprocity hybrid boundary node method for nonlinear problems

In this paper, a boundary type meshless method of dual reciprocity hybrid boundary node method (DHBNM) is proposed to solve complicate Poisson type linear and nonlinear problems. Firstly, the solutions are divided into the complementary solutions related to homogeneous equation and the particular solutions solved by nonhomogeneous terms, for the latter, they are approximated by the radial basis function interpolation based on dual reciprocity method, and the complementary solutions are obtained based on simple Poisson's equation by hybrid boundary node method, by which a simple fundamental solution of the Laplacian operator is employed instead of some other complicated ones; then a function of field functions and their derivatives on any point can be easily obtained, employing the concept of the analog equation of Katsikadelis, the field functions and their derivatives can be expressed as the function of unknown series of coefficients, and a series of nonlinear equivalent equations can be established by collocating the original governing equation at discrete points in the interior and on boundary of the domain. As a result, a new meshless method of dual reciprocity hybrid boundary node method is proposed to solve nonlinear Poisson type problems, because of the usage of those techniques, the boundary type meshless properties can be kept for any type of nonlinear equations. Different types of classical nonlinear problems are presented to validate the effectiveness and the accuracy of the present method.

Density function of random differential equations via finite difference schemes: a theoretical analysis of a random diffusion-reaction Poisson-type problem

ABSTRACT A computational approach to approximate the probability density function of random differential equations is based on transformation of random variables and finite difference schemes. The theoretical analysis of this computational method has not been performed in the extant literature. In this paper, we deal with a particular random differential equation: a random diffusion-reaction Poisson-type problem of the form , , with boundary conditions , . Here, α, A and B are random variables and is a stochastic process. The term is a stochastic process that solves the random problem in the sample path sense. Via a finite difference scheme, we approximate with a sequence of stochastic processes in both the almost sure and senses. This allows us to find mild conditions under which the probability density function of can be approximated. Illustrative examples are included.

Radial ground state sign-changing solutions for a class of asymptotically cubic or super-cubic Schrödinger–Poisson type problems

This paper is dedicated to studying the following Schrodinger–Poisson system $$\begin{aligned} \left\{ \begin{array}{ll} -\triangle u+V(|x|)u+\lambda \phi u=f(|x|,u), &{}\quad x\in \mathbb {R}^3,\\ -\triangle \phi = u^2,&{}\quad x\in \mathbb {R}^3, \end{array} \right. \end{aligned}$$ where $$\lambda $$ is a positive parameter, $$V\in {\mathcal {C}}(\mathbb {R}^{3}, (0,\infty ))$$ and $$f\in {\mathcal {C}}({\mathbb {R}}^{3}\times \mathbb {R}, \mathbb {R})$$ . Using weaker conditions $$\lim _{|t|\rightarrow \infty }\frac{\int _0^tf(x, s){\mathrm {d}}s}{|t|^3}=\infty $$ uniformly in $$x\in \mathbb {R}^3$$ , and $$\begin{aligned} \left[ \frac{f(x,\tau )}{\tau ^3}-\frac{f(x,t\tau )}{(t\tau )^3}\right] {\mathrm {sign}}(1-t) +\theta _0V(x)\frac{|1-t^2|}{(t\tau )^2}\ge 0, \quad \forall \;\; x\in \mathbb {R}^3,\ t>0, \;\; \tau \ne 0 \end{aligned}$$ with a constant $$\theta _0\in (0,1)$$ , instead of the usual super-cubic condition $$\lim _{|t|\rightarrow \infty }\frac{\int _0^tf(x, s){\mathrm {d}}s}{|t|^4}=\infty $$ uniformly in $$x\in \mathbb {R}^3$$ , and the Nehari type monotonic condition on $$f(x,t)/|t|^3$$ , we establish the existence of one radial ground state sign-changing solution $$u_\lambda $$ with precisely two nodal domains. Under the same conditions, we also prove that the energy of any radial sign-changing solution is strictly larger than two times the least energy; and give a convergence property of $$u_\lambda $$ as $$\lambda \searrow 0$$ . Our result unifies both asymptotically cubic and super-cubic cases, which extends the existing ones in the literature.

A parallel multigrid solver for incompressible flows on computing architectures with accelerators

An efficient parallel multigrid pressure correction algorithm is proposed for the solution of the incompressible Navier–Stokes equations on computing architectures with acceleration devices. The pressure correction procedure is based on the numerical solution of a Poisson-type problem, which is discretized using a fourth-order finite difference compact scheme. Since this is the most time-consuming part of the solver, we propose a parallel pressure correction algorithm using an iterative method based on a block cyclic reduction solution method combined with a multigrid technique. The grid points are numbered with respect to the red–black ordering scheme for the parallel Gauss–Seidel smoother. These parallelization techniques allow the execution of the entire simulation computations on the acceleration device, minimizing memory communication costs. The realization is developed using the OpenACC API, and the numerical method is demonstrated for the solution of two classical incompressible flow test problems. The first is the two-dimensional lid-driven cavity problem over equal mesh sizes while the other is the Stokes boundary layer, which is a decent benchmark problem for unequal mesh spacing. The effect of several multigrid components on modern and legacy acceleration architectures is examined. Eventually the performance investigation demonstrates that the proposed parallel multigrid solver achieves an acceleration of more than 10\(\times \) over the sequential solver and more than 4\(\times \) over multi-core CPU only realizations for all tested accelerators.

Generalized finite element methods for quadratic eigenvalue problems

We consider a large-scale quadratic eigenvalue problem (QEP), formulated using P1 finite elements on a fine scale reference mesh. This model describes damped vibrations in a structural mechanical system. In particular we focus on problems with rapid material data variation, e.g., composite materials. We construct a low dimensional generalized finite element (GFE) space based on the localized orthogonal decomposition (LOD) technique. The construction involves the (parallel) solution of independent localized linear Poisson-type problems. The GFE space is used to compress the large-scale algebraic QEP to a much smaller one with a similar modeling accuracy. The small scale QEP can then be solved by standard techniques at a significantly reduced computational cost. We prove convergence with rate for the proposed method and numerical experiments confirm our theoretical findings.

Ground state sign-changing solutions for a class of Schrödinger–Poisson type problems in $${\mathbb{R}^{3}}$$ R 3

This paper is dedicated to studying the following Schrodinger–Poisson system $${\left\{ \begin{array}{ll}-\triangle u+V(x)u+\lambda\phi u=K(x)f(u),& \quad x\in \mathbb{R}^{3},\\-\triangle\phi= u^2,\quad x\in \mathbb{R}^{3},\end{array}\right.}$$ where V,K are positive continuous potentials, f is a continuous function and \({\lambda}\) is a positive parameter. We develop a direct approach to establish the existence of one ground state sign-changing solution \({u_\lambda}\) with precisely two nodal domains, by introducing a weaker condition that there exists \({\theta_0\in (0,1)}\) such that $$K(x)\left[\frac{f(\tau)}{\tau^3}-\frac{f(t\tau)}{(t\tau)^3} \right]\mathrm{sign}(1-t)+\theta_0V(x)\frac{|1-t^2|}{(t\tau)^2} \geq 0, \quad \forall x \in\mathbb{R}^3, t > 0, \tau\ne 0$$ than the usual increasing condition on \({f(t)/|t|^3}\). Under the above condition, we also prove that the energy of any sign-changing solution is strictly larger than two times the least energy, and give a convergence property of \({u_\lambda}\) as \({\lambda\searrow 0}\).

One-stage Method of Fundamental and Particular Solutions (MFS-MPS) for the steady Navier–Stokes equations in a lid-driven cavity

The coupled nonlinear steady state Navier–Stokes (N–S) equations in the stream function–vorticity form for a lid-driven cavity are solved by a one-stage Method of Fundamental Solutions (MFS) and the Method of Particular Solutions (MPS). This method has been earlier used for linear Poisson-type problems and has not been applied to coupled nonlinear equations. In this method the steady state N–S equations are first put in the form of two nonlinearly coupled Poisson equations and the solution is sought as the sum of their respective homogeneous and particular solutions. The homogeneous solution is obtained using the MFS and the particular solution is found with the help of Radial Basis Functions (RBFs). Both the operations are accomplished in a single stage. The nonlinear coupling of the N–S equations is tackled by iteration and successive relaxation. We find that the method is easy and effective when compared with the boundary element method (BEM) or the two-stage MFS-MPS, due to its meshless, singular integration free qualities and the single stage operation. The results are obtained for the moderate Reynolds numbers by varying the relaxation parameter. The convergence of MFS-MPS scheme for the present nonlinear problem is numerically demonstrated.

Finite element network approximation of conductivity in particle composites

A new finite element method computes conductivity in some unstructured particle-reinforced composite material. The 2-phase material under consideration is composed of a poorly conducting matrix material filled by highly conducting circular inclusions which are randomly dispersed. The mathematical model is a Poisson-type problem with discontinuous coefficients. The discontinuities are huge in contrast and quantity. The proposed method generalizes classical continuous piecewise affine finite elements to special computational meshes which encode the particles in a network structure. Important geometric parameters such as the volume fraction are preserved exactly. The computational complexity of the method is (almost) proportional to the number of inclusions. This is minimal in the sense that the representation of the underlying geometry via the positions and radii of the inclusions is of the same complexity. The discretization error is proportional to the distance of neighboring inclusions and independent of the conductivity contrast in the medium.

The AL Basis for the Solution of Elliptic Problems in Heterogeneous Media

In this paper, we will show that, for elliptic problems in heterogeneous media, there exists a local (generalized) finite element basis (AL basis) consisting of $O\big( \big( \log\frac{1}{H}\big) ^{d+1}\big)$ basis functions per nodal point such that the convergence rates of the classical finite element method for Poisson-type problems are preserved. Here H denotes the mesh width of the finite element mesh and d is the spatial dimension. We provide several numerical examples beyond our theory, where even $O(1)$ basis functions per nodal point are sufficient to preserve the convergence rates.

Solving the nonlinear Poisson-type problems with F-Trefftz hybrid finite element model

A hybrid finite element model based on F-Trefftz kernels (fundamental solutions) is formulated for analyzing Dirichlet problems associated with two-dimensional nonlinear Poisson-type equations including nonlinear Poisson–Boltzmann equation and diffusion–reaction equation. The nonlinear force term in the Poisson-type equation is frozen by introducing the imaginary terms at each Picard iteration step, and then the induced Poisson problem is solved by the present hybrid finite element model involving element boundary integrals only, coupling with the particular solution method with radial basis function interpolation. The numerical accuracy of the present method is investigated by numerical experiments for problems with complex geometry and various nonlinear force functions.

An efficient method for solving elliptic boundary element problems with application to the tokamak vacuum problem

A method for regularizing ill-posed Neumann Poisson-type problems based on applying operator transformations is presented. This method can be implemented in the context of the finite element method to compute the solution to inhomogeneous Neumann boundary conditions; it allows to overcome cases where the Neumann problem otherwise admits an infinite number of solutions. As a test application, we solve the Grad–Shafranov boundary problem in a toroidally symmetric geometry. Solving the regularized Neumann response problem is found to be several orders of magnitudes more efficient than solving the Dirichlet problem, which makes the approach competitive with the boundary element method without the need to derive a Green function. In the context of the boundary element method, the operator transformation technique can also be applied to obtain the response of the Grad–Shafranov equation from the toroidal Laplace n=1 response matrix using a simple matrix transformation.

A collocation and Cartesian grid methods using new radial basis function to solve class of partial differential equations

In the present paper, a class of partial differential equation represented by Poisson's type problems are solved using a proposed Cartesian grid method and a collocation technique using a new radial basis function. The advantage of using this new radial basis function represented by overcoming singularity from the diagonal elements when thin plate radial basis function is used. The new function is a combination of both multiquadric and thin plate radial basis functions. The new radial basis function contains a control parameter ϵ, that takes one when evaluating the singular elements and equals zero elsewhere. Collocation of the approximate solution of the potential over the governing and boundary condition equations leads to a double linear system of equations. A proposed algebraic procedure is then developed to solve the double system. Examples of Poisson and Helmholtz equations are solved and the present results are compared with the their analytical solutions. A good agreement with analytical results is achieved.

Partitioned formulation of internal and gravity waves interacting with flexible structures

This paper presents a partitioned modeling of internal and gravity fluid waves that interact with flexible structures. The governing interaction model consists of three completely partitioned entities: fluid model, structural model, and interface model that acts as an internal constraint on the fluid–structure interface boundary. Thus, the proposed partitioned multi-physics modeling can employ two completely modular fluid and structure software modules plus an interface solver, hence amenable to partitioned solution algorithms. The interface discretization can exploit the nonmatching interface algorithm previously developed via the method of localized Lagrange multipliers. Also noted is that the present fluid model can make use of widely available finite element software for standard Poisson-type problems.

Guidelines for Poisson Solvers on Irregular Domains with Dirichlet Boundary Conditions Using the Ghost Fluid Method

We consider the variable coefficient Poisson equation with Dirichlet boundary conditions on irregular domains. We present numerical evidence for the accuracy of the solution and its gradients for different treatments at the interface using the Ghost Fluid Method for Poisson problems of Gibou et al. (J. Comput. Phys. 176:205---227, 2002; 202:577---601, 2005). This paper is therefore intended as a guide for those interested in using the GFM for Poisson-type problems (and by consequence diffusion-like problems and Stefan-type problems) by providing the pros and cons of the different choices for defining the ghost values and locating the interface. We found that in order to obtain second-order-accurate gradients, both a quadratic (or higher order) extrapolation for defining the ghost values and a quadratic (or higher order) interpolation for finding the interface location are required. In the case where the ghost values are defined by a linear extrapolation, the gradients of the solution converge slowly (at most first order in average) and the convergence rate oscillates, even when the interface location is defined by a quadratic interpolation. The same conclusions hold true for the combination of a quadratic extrapolation for the ghost cells and a linear interpolation. The solution is second-order accurate in all cases. Defining the ghost values with quadratic extrapolations leads to a non-symmetric linear system with a worse conditioning than that of the linear extrapolation case, for which the linear system is symmetric and better conditioned. We conclude that for problems where only the solution matters, the method described by Gibou, F., Fedkiw, R., Cheng, L.-T. and Kang, M. in (J. Comput. Phys. 176:205---227, 2002) is advantageous since the linear system that needs to be inverted is symmetric. In problems where the solution gradient is needed, such as in Stefan-type problems, higher order extrapolation schemes as described by Gibou, F. and Fedkiw, R. in (J. Comput. Phys. 202:577---601, 2005) are desirable.

Chebyshev tau matrix method for Poisson-type equations in irregular domain

In this paper, a new meshless method, Chebyshev tau matrix method (CTMM) is researched. The matrix representations for the differentiation and multiplication of Chebyshev expansions make CTMM easy to implement. Problems with curve boundary can be efficiently treated by CTMM. Poisson-type problems, including standard Poisson problems, Helmholtz problems, problems with variable coefficients and nonlinear problems are computed. Some numerical experiments are implemented to verify the efficiency of CTMM, and numerical results are in good agreement with the analytical one. It appears that CTMM is very effective for Poisson-type problems in irregular domains.