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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How can I find a solution to this fourth order PDE?
I am trying to find a solution to the following PDE:
$$
\nabla^2(\nabla^2\Psi-M)-\epsilon\nabla(\nabla\Psi M)=0
$$
Where $\Psi(x,y,z)$ and $M(x,y,z)$ are scalar fields in 3D space and $M(x,y,z)\geq 0$ ...
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Maximum Principle for Poisson’s Equation
I understand how to prove the maximum principle for $u_{xx}+u_{yy}=0$, but how does this extend to a maximum principle for the equation $u_{xx}+u_{yy}=f$?
I believe this is called Poisson’s equation. ...
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How to prove the uniqueness and stability of solution to Poisson equation using the maximum principle?
I came across this problem in an exam. Given a Poisson equation on a bounded region $\Omega$,
$$\begin{cases}
-\Delta u=f& u\in\Omega\\
\alpha u+\beta\dfrac{\partial u}{\partial n}=g&u\in\...
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answer
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Representation of the analytical solution to the Poisson equation?
I have the following 1-dimensional Poisson's problem with the corresponding boundary conditions:
$$
u_{xx} = -f(x) ; \quad u(1) = g, \quad -u_x(0) = h
$$
on a open domain
$$
\Omega = (0,1)
$$
Is it ...
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Regularity on the Poisson equation on bounded domains
This is a follow-up question to Regularity of solution for Poisson equation on bounded domains.
We consider the Poisson equation $-\Delta u=f$ in $\Omega$ and $u=\varphi$ on $\partial\Omega$.
...
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answer
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weak variational formulation of Poisson equation with some restrictions.
The variational formulation of Poisson equation
$$
\begin{cases}
-\Delta u = f & \text{in } \Omega \\
u = 0 & \text{on } \partial \Omega
\end{cases}
$$
is that: Find $u\in H_0^1(\Omega)$, such ...
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Solution theory of Poisson problem with complex zero-order term
We assume that $\Omega \subseteq \mathbb R^n$ is a bounded open set with smooth boundary and study the PDE
$$
-\Delta u + \alpha u = f
$$
with, say, zero Dirichlet boundary conditions along $\partial\...
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Regularity of solution for Poisson equation on bounded domains
Let $\Omega\subset \mathbb{R}^n$ be a bounded domain and consider $-\Delta u=f$ in $\Omega$, $u=\varphi$ on $\partial\Omega$.
For balls with some radius the following result is known: Let $\varphi \...
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How to prove this Sobolev inequality?
$$
\Vert uv\Vert_{H^m} \leq C\sum_{j=2}^m \Vert u\Vert_{H_j} \Vert v\Vert_{H^{m-j+2}}, m\geq 2$$
This inequality from
Auzinger, Winfried; Kassebacher, Thomas; Koch, Othmar; Thalhammer, Mechthild, ...
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Poisson equation 2D with dirichlet BC. What is the expression of u(x,y) in terms of a, b, c, x and y using variable separation and Fourier method? [duplicate]
The 2D Poisson equation with Dirichlet boundary conditions of zero and a constant source term $c$ can be written as:
$$
\nabla^2 u(x, y) = c, \quad \text{for } (x, y) \in \Omega \\
$$
where $\Omega$ ...
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Test functions in the weak form of Poisson equation
Consider the Poisson equation with zero Dirichlet boundary condition: Find $u \in C^2 (\Omega)\cap C(\overline{\Omega})$ s.t.
$$\begin{cases}
-\Delta u=f&in\ \Omega {,}\\
u\mid _{\partial \Omega }=...
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integral over Green's function of a $d$-Ball
I'm studying for an exam on PDE's and I'm stuck trying to solve the following problem.
This is in context of the following boundary value problem:
\begin{align}
\begin{cases}
-\Delta u \equiv 0,&\...
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A priori estimate of solution of Poisson equation
I'm studying for an exam on PDE's and came across the below problem. I found an approach for solving it, but am missing a last argument. Can you help me conclude my proof?
This is related to this ...
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answer
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2D Poisson's Equation within a disk
As a vector calculus exercise, I want to solve following PDE $$\nabla^2 u = r^2$$ on a 2D domain $D = \{(r,\theta) : 1 \leq r \leq 2\}$ plus a condition that $u=1$ whenever $r=1$ and $r=2$.
There was ...
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Characterization of the Poisson distribution
Let $Z$ be a $[0, \infty)$-valued random variable satisfying $\lambda E[g(Z+1)] = E[Zg(Z)]$ for all indicator functions $g$ of Borel subsets of $[0, \infty)$.
Prove that $\mathcal{L}(Z) = Poisson(\...
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answer
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General compatibility condition for pure Neumann problem
It is known than for a second order elliptic boundary-value problem with pure Neumann conditions, a certain compatibility condition between the data must be satisfied.
For example, in the case of the ...
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answer
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Poisson problem with discontinuous coefficient
The task is to solve for the exact weak solution to
$$ \begin{cases} -\partial_x (k(x) \partial_x u(x)) = 0 & \text{for } x \in (-1,1),\\
u = 0 & \text{at } x = -1,\\
u = 3 & \...
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How to numerically solve a "quasi-continuous" set of Poisson differential equations?
In the context of some research in Theoretical Chemistry regarding the magnetic response of the electrons in a molecule I came across the following problem:
I have numerical data of a smooth and ...
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votes
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answer
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Legendre polynomial as basis for finite element method
When people say use Legendre polynomial as basis polynomial in fem, are they using the polynomials themselves or the integral of those polynomials?
I'm asking this because I have this poisson equation:...
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References for the newtonian potential
I am looking for reference for the newtonian potential, the only one I've found is chapter 4 of Elliptic partial differential equations of second order by Gilbarg and Trudinger but I find it very ...
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Does distributional Laplacian imply regularity
I come up with the following questions when considering the abstract setting of mixed finite elements:
If $u, f\in L^2(\Omega)$ and for any $\phi\in C_c^{\infty}(\Omega)$,
$$\int_\Omega u\Delta \phi=\...
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Estimate of normal derivative of a function $u$ which solves Poisson´s equation. [duplicate]
Let $\Omega$ be a bounded $C^1$ domain satisfying the exterior sphere condition at every boundary point and $f$ be a bounded continuous function in $\Omega$. Suppose $u \in C^2(\Omega) \cap C^1(\...
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Poisson equation, spectral method and Hermitian symmetry of DFT
Let's say I have 2D rectangular grid $N \times M$ with the real values values $f_{nm}$.
I would like to solve 2D Poisson equation
$$\Delta u = f \,\,\, in \,\,\, [0,A] \times [0,B].$$
with periodic ...
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answers
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Solving finite element equations element-by-element without storing the global K matrix
I am looking for articles or books that explain how to solve the finite element problems element-by-element without having to assemble and construct the full global stiffness matrix. I find it hard to ...
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answer
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Proof of Decay Estimate for Poisson Equation
In my class we learnt about the decay estimate of the Poisson equation. The statement is:
However, I do not understand the proof given in the notes. The key part of the theorem since to be that $u(x) ...
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Positive solution of steady-state temperature with space-dependent conductivity
In Positive Solution of Poisson Equation it has been shown that $u(x)$ solution to the Poisson equation remains non-negative if $f(x) \geq 0$ and $u=0$ on $\Gamma_D$ based on the fact that the trace ...
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fixing the error to be the same in every point in grid
maximum norm of the error is smaller than 2% at every point in the grid. That is, we
measure the relative error against the true solution u
and wish to choose the largest $h_{t}$ and $h_{x}$ (say, up ...
user1119555
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answer
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Positive Solution of Poisson Equation
Given the Poisson's equation
\begin{equation}
-\nabla^2 u = f \quad \mathrm{in} \ \Omega
\end{equation}
\begin{equation}
u=0 \quad \mathrm{on} \ \Gamma_D, \quad \frac{\partial u}{\partial n} = 0 \quad ...
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Harnack inequality on upper half ball
Define the upper half ball in $\mathbb{R}^n$ to be the set $B_r(0)^+ = B_r(0) \cap \{y_{n}>0\} .$ Denote for $y = (y',y_n)\in B_r(0)^+$, $$\tilde{y} = (y',-y_{n}).$$ Then for a harmonic function $...
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Radial solution Poisson Equation
I am looking for a radial solution on $B_1(0) \subset \mathbb{R}^2$.
$$\Delta v=v''(r)+ {1 \over r}v'(r)=r-1$$
I already know that $v(r)=c_1 \ln(r)+c_2$ is a solution for $\Delta v=0$ and $v(1)=0$.
...
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Solving a linear system of equations (finite difference method). Treatment of ghost cells?
I am trying to solve the Poisson equation on a 3D grid:
$\dfrac{\partial^2 f}{\partial x^2}+\dfrac{\partial^2 f}{\partial y^2}+\dfrac{\partial^2 f}{\partial z^2}=C$
The equation is discretized using ...
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vote
1
answer
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Pressure values at the boundary in Navier Stokes equations
An interesting and important question: what boundary conditions should be used for the pressure field in Navier-Stokes system of equations?
I'm currently trying to find a way to derive the pressure ...
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Radial poisson equation in spherical polar coordinates
(This question may belong on Physics. I put in on Mathematics because I saw more similar questions here.)
I have a seperable density function that is expressed in spherical polar coordinates as:
$$n_{...
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votes
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answers
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Showing that the Poisson Kernel for the unit disk over the interval $-\pi$ to $\pi$ is equal to 1 for $0<r<1$
I know the Poisson Kernel is $$P(r,\xi)=\frac{1}{2\pi}\left(1+2\sum^{\infty}_{n=1}r^n\cos(n\xi)\right)$$
so $$U(r,\theta)=\int_{-\pi}^{\pi}P(r,\xi-\theta)f(\xi)d\xi$$
The Poisson's integral formula is$...
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Apply the Poisson formula to obtain the expression of a harmonic function
For a harmonic function $u$ in $\mathbb{D}\subset\mathbb{C}$, with boundary value
$$
u(e^{i\theta})=-\log\dfrac{1}{|\xi-e^{iN\theta}|^2}
$$
where $|\xi|>1$ is a fixed complex number,
theoretically ...
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Solving the Poisson's equation using intergal transform method
Consider the following Poisson's equation;
$${\nabla}^2\phi=-f(\mathbf{r})$$
How can I solve for $\phi$ using integral transform?
I tried beginning with inverse fourier transform but just after I ...
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answers
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Formula for the inverse of Laplacian plus constant in a ball\semiball
Let $\lambda_1$ be the first positive eigenvalue of the following problem in the unit semiball $\mathbb{B}_+^n = \{x \in \mathbb{R}^n : \vert x \vert \leq 1 \text{ and } x_n \geq 0 \}$:
\begin{cases}
\...
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Problems with the direct solution of Poisson equation in spherical coordinates
I have a charge density $\rho(\vec{r})$ which is given as an expansion of spherical harmonics $Y_{l}^{m}(\hat{r})$.
$$\rho(\vec{r}) = \sum_l^{l_{max}} \sum_{m = -l}^{l} \rho_{l}(r) Y_{l}^{m}(\hat{r}) ...
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Successive over-relaxation method for solving partial differential equations and pure Neumann boundary conditions
I've recently posted this question on Computational Science Stack Exchange in hopes of getting some insights on how to deal with ill-conditioned 2D Poisson partial differential equation (PDE) which ...
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An $n-$dimensional Poisson Scheme
We know about two and three-dimension Poisson equations.
Question: Is there any literature done on generalization of Poisson equation in $\mathbb{R}^{n}$?
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Bounds for the $H^1$ norm of the solution to Poisson's equation
Problem
Let $\Omega \subset \mathbb{R}^N$ be a bounded domain with a Lipschitz continuous boundary $\partial \Omega$; additionally, let $f \in L^2(\Omega)$ and $g \in H^{1/2}(\partial \Omega)$ be ...
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Elliptic PDE: FEM gives smaller error than FDM in the same mesh?
I came from this question which compares both in a general way, but it doesn't mention about the numerical error.
I've learned that FEM uses Galerkin's method to minimize the integral of the residual ...
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Does $\Delta u=f$ if $\{u_i\}$ and $\{f_i\}$ are smooth functions such that $u_i\to u$ in $W^{k+2,p}$, $f_i\to f$ in $W^{k,p}$, and $\Delta u_i=f_i$?
Question: Let $\{u_i\}$ and $\{f_i\}$ be two sequences of functions in $C^\infty$, and suppose that $u_i\to u$ in the Sobolev space $W^{k+2,p}$ and that $f_i\to f$ in $W^{k,p}$. If each $u_i$ solves ...
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Pde equation with p-powers
I would like to find some references and naming about the following class of PDEs
$$
\sum_i\partial^p_{x_i}u(\mathbf{x})=0,\quad \mathbf{x}=(x_1,x_2,\dots,x_n),
$$
for $p\geq1$.
In the case $p=1$, (...
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Dirichlet problem for Poisson equality
Let $D = \lbrace\left(x_1,x_2,x_3\right) \in \mathbb{R}^3 : x_3 \gt 0\rbrace$. Solve Dirichlet problem for Poisson equlity $-\Delta=f$ in $D$, $u = g$ on $\Gamma$ in space of limited functions, if:
(a)...
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Weak formulation of Poisson's equation: is the bilinear form coercive?
Let us consider the Poisson's equation
$$-\nabla ^{2}u=f,$$
on a domain $\Omega \subset {\mathbb R}^{d}$ with $u=0$ on its boundary. Use the $L^{2}$-scalar product
$$\langle u,v\rangle =\int _{\Omega ...
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answer
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Conditions for the existence of global, weak solutions to Poisson's Equation on $\mathbb{R}^3$
I have been thinking a lot lately about Poisson's equation:
$$
\nabla^2 \phi = f
$$
where $f: \mathbb{R}^3 \rightarrow \mathbb{R}$ is given, and we solve for $\phi: \mathbb{R}^3 \rightarrow \mathbb{R}$...
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Solving poisson equation in polar coordinates for a separable potential.
I would like to get a step-by-step solution of the Poisson equation (in polar coordinates) below
$$\nabla^2\psi(r, \phi) = 2 k(r, \phi)$$
where $\psi(r, \phi)$ is seperable (i.e. $\psi(r, \phi) = f(r)...
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Electrostatic potential of a charge in a 3-torus
Consider the 3-torus which arises from taking $R^3$ and identifying two points $x\equiv x+nL$ whenever $n$ is a vector with integer components. I'm curious about finding the electrostatic potential ...
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Explicit solution to Poisson equation on torus
I am studying the Vlasov-Poisson system on a torus in three dimensions, $\mathbb{T}^3$. I consider a constant mass density $\rho = C$. What is a solution to the Poisson equation $\Delta U = 4\pi\rho $ ...
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