# Poisson

Algorithms for solving the 2D and 3D Poisson’s equation

## Problem

Let $u\in\mathbb{R}$, solve:

$ -\displaystyle{\Delta u = f} $

With $f$ an external force.

## Variational form

Let $\Omega\in\mathbb{R}^n$, $2\leq n\leq3$. Let $u, v\in H^1(\Omega)$. The variational form reads as follows:

$ \displaystyle{-\int_{\Omega}{\Delta u v} - \int_{\Omega}{f v} = 0} $

Using the Green formula:

$ \displaystyle{\int_{\Omega}{\nabla u \cdot \nabla v} - \int_{\Omega}{f v} = 0} $

## Algorithms

### 2D

Poisson’s equation on a square.

Result |
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### 3D

Poisson’s equation on a cube.

Result |
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## Validation

### 2D

Let $u_e$ be the analytical solution on $\Omega=[0,1]^2$:

$ \displaystyle{ u_e = x(1-x)y(1-y)e^{x-y} } $

We get, for homogenous Dirichlet conditions on $\partial\Omega$:

$ \displaystyle{ f = -2x(y-1)(y-2x+xy+2)e^{x-y} } $

and the following convergence curve:

Convergence curve |
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The slope obtained with this algorithm is equal to $3.0235$, which corresponds to the theoretical value of $3$.

The complete validation script is available here

## Authors

Author: Simon Garnotel