# 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)

447 questions
Filter by
Sorted by
Tagged with
10 views

### 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 $M(x,y,z)\geq 0$ ...
21 views

### 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. ...
21 views

25 views

### 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,&\...
26 views

### 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 ...
50 views

### 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 ...
1 vote
43 views

221 views

### 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 ...
28 views

### 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 ...
1 vote
32 views

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 $u(x) ... 0 votes 0 answers 14 views ### Positive solution of steady-state temperature with space-dependent conductivity In Positive Solution of Poisson Equation it has been shown that$u(x)$solution to the Poisson equation remains non-negative if$f(x) \geq 0$and$u=0$on$\Gamma_D$based on the fact that the trace ... 0 votes 0 answers 15 views ### fixing the error to be the same in every point in grid maximum norm of the error is smaller than 2% at every point in the grid. That is, we measure the relative error against the true solution u and wish to choose the largest$h_{t}$and$h_{x}$(say, up ... 3 votes 1 answer 78 views ### Positive Solution of Poisson Equation Given the Poisson's equation \begin{equation} -\nabla^2 u = f \quad \mathrm{in} \ \Omega \end{equation} \begin{equation} u=0 \quad \mathrm{on} \ \Gamma_D, \quad \frac{\partial u}{\partial n} = 0 \quad ... 3 votes 0 answers 168 views ### Harnack inequality on upper half ball Define the upper half ball in$\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$...
1 vote
41 views

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 $v(1)=0$. ...
60 views

### 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 ...
1 vote
197 views

### 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 ...
1 vote
59 views

102 views

### 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 ...
17 views

### 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 $\mathbb{R}^{n}$?
1 vote
128 views

### 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 ...
30 views

### 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 ...
1 vote
41 views

### 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 ...
17 views

### 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 $p=1$, (...
1 vote
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)...