# Questions tagged [poissons-equation]

In mathematics, Poisson's equation is a partial differential equation of elliptic type with broad utility in electrostatics, mechanical engineering and theoretical physics. (Def: https://en.wikipedia.org/wiki/Poisson%27s_equation)

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### Solving 1D Poisson equation using finite element method and understanding the Galerkin orthogonality

Let's consider the following test problem $$u'' = 12x^2 - 36x + 18 \qquad u(0) = u(3) = 0$$ Analytical solution is $$u(x) = (x-3)^2 x^2$$ I'm solving this using the finite element method, ...
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### The laplacian of the solution of FastPoisson is off by the mean of the RHS

I'm trying to implement a FastPoisson solver to solve the (two-dimensional) equation $\nabla^2u=x$ with Neumann boundary conditions. The algorithm goes like this ($N_x$ and $N_y$ are the width and ...
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### Planet Zog: Gravity of a Sphere with Spherical Deletions

Years ago I was given this problem to do. I couldn't manage it at the time but was given the broad strokes of the solution. I came across it again recently and decided to have a go. Could someone ...
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### FFT solve Poisson, why amplitude not match

I'm testing the FFT method, use a Poission equation $\Delta u = -16\pi\sin(4\pi x)$ The analytical solution should be $u = \sin(4\pi x)$ My result is more or less similarly, however the amplitude ...
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### Why is $a$ not coercive when it is defined on $H^1(\Omega)$?

Given the Poisson's equation with homogeneous Neumann boundary conditions and the associated bilinear form $$a(u,v) = \int_{\Omega}\nabla u \cdot \nabla v \, dx$$ on $H^1(\Omega)$, why is $a$ not ...
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### PDE-Poisson Equation $u_{xx}+u_{yy}=-1$?? [closed]

I want to solve the partial differential equation $$u_{xx}+u_{yy}=-1$$ in the region $0<x<1, y>0$ subject to the boundary conditions $u(0,y)=0, u(1,y)=1$ and $u(x,0)=0$.
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### Existence and uniqueness of solenoidal decomposition $f=f^{\text{s}}+\nabla\phi$ for a vector field $f\colon\mathbb{R}^n\rightarrow\mathbb{R}^n$?

I am interested in the existence of a global solution to the Poisson equation $$\Delta\phi=g \quad \text{in} \ \mathbb{R}^n$$ such that $\phi\rightarrow 0$ when $|x|\rightarrow\infty$. My motivation ...
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### Estimate solution of Poisson equation on unit ball

Consider the following boundary value problem where $U=\{x \in \mathbb{R}^3 \mid |x|<1\}$ and $g$ is some nice bounded function, $$\Delta u = g ~~~ \text{on}~U\\ u=0 ~~~\text{on} ~\partial U.$$ ...
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### Probability of a Poisson variable being greater than k others

If one has multiple Poisson random variables (with potentially different lambda parameter), how can one theoretically find the probability that one of those variables is greater than the others? That ...
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### Determining green's function inside a circle for Poisson/Laplacean PDE

In the 2D circle of radius $A$, how does one determine the Green's function for the 2D Poisson Equation? I recognize that you have to use a reflection technique so that the function is 0 on the ...
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### Poisson Equation Solution in Python

I want to solve the Poisson Equation for potential, $V(r)$ at $\delta$. But I am not sure how to do it. I have values of charge density, $\rho(r)$ at discrete values of $r$ stored in an array. Values ...
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### Variational formulation Poisson equation (1d FEM)

I have the same question of this one $\bullet$ In the answer I've seen that a "Lifting" function is used, is it to have a formulation where the test functions are in the same space of the solution? ...
Considering the Poisson Equation $-u_{xx} - u_{yy}=1$ in the unit square $\Omega = [0,1] \times [0,1]$, with homogeneous Dirichlet boundary conditions, I am required to compute the second derivative ...
$u$ is harmonic in $\mathbb{R}^n_+$ and $u=0$ on the boundary. I wish to extend $u$ to a harmonic function on $\mathbb{R}^n$. Suppose I defined $u(x_1,...,x_{n-1},x_n)=u(x_1,...,x_{n-1},-x_n)$ for \$...