# 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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### What is wrong with using the $H^1_0$ inner product here?

This question is a problem I am toying with. Consider the Poisson equation, $$-\Delta u =f \ \text{in} \ \Omega \times (0,\infty).$$ $$u=0 \ \text{on} \ \partial \Omega \times (0,\infty)$$ Suppose the ...
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### Solving the Radially Symmetric Poisson Equation with Exponential Source Term

I want to solve the poisson equation $$-\Delta u(\mathbf{x})=\rho(\mathbf{x})=\frac{e^{-|\mathbf{x}|^2}}{|\mathbf{x}|^2-1}$$ The problem want me to use the fundamental solution of laplace operator, ...
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### Fourier transform of Poisson's equation with periodic shifted boundary conditions

I am trying to solve following 2D Poisson's equation numerically: $$\Delta \Phi = \rho$$ When $\rho$ and $\Phi$ are periodic in both directions, this can be solved straightforward thanks to Fourier ...
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### Regularity of solutions to Poisson's equation on part of the boundary

Let $\Omega \subset \mathbb{R}^n$ be an open and bounded set and suppose $\partial \Omega$ is smooth on a relatively open subset $\Gamma \subset \partial \Omega$. Consider a weak solution of the ...
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### Solving Poisson equation with separate variable method

Here's a Poisson equation, $D$ is a unit square i.e. $[0,1]\times[0,1]$. \begin{equation*} \begin{aligned} \Delta u(x,y) &= \varphi(x,y) \quad (x,y)\in D \\ u(x,y) &= 0 \quad (x,y)\in\partial ...
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### Showing uniqueness of solution to a non-linear Poisson problem

I'm trying to prove that a non-linear Poisson problem has a unique solution. The context is the following: Let $\Omega \subset \mathbb{R}^n$ be a bounded open subset of class $C^2$. Consider the ...
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### Schauder estimates on boundary

I recently studied the Schauder estimates with the boundary and checked the wiki page. The following is the link (the boundary estimate is on the bottom part): https://en.wikipedia.org/wiki/...
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### Unique solvability of weak Poisson equation with Neumann boundary condition

I'm studying a Poisson BVP: Find $u \in W^{1,p}(\Omega)$ such that $$\int_{\Omega} \nabla u \cdot \nabla v = F(v) \quad \forall v \in W^{1,p'}(\Omega),$$ where $p$ and $p'$ are Hölder conjugates and ...
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### Why do mathematicians use BDM or Hdiv finite elements to solve the Mixed Poisson partial differential equation

I am still new to finite element methods, and I was looking at some tutorials on a specific formulation of the Poisson equation that introduces an additional variable. Some of the tutorials call this ...
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Say we have a Poisson problem: $\nabla^2 \varphi = S$ As a boundary value problem, it requires the definition of boundary conditions on all surfaces of the domain. If we assume there is a vector field ...