Poisson equation

Verification of the solution of the Poisson equation (Laplacian operator) More...

Topics
	1D Poisson equation with Dirichlet boundary conditions
	1D Poisson test cases with Dirichlet boundary conditions
	1D Poisson equation with Robin boundary conditions
	1D Poisson test cases with Robin boundary conditions
	2D/3D Poisson equation with cylinder/sphere immersed boundaries
	This test case suite focuses on the immersed boundary methods applied to the Poisson equation.
	2D/3D Poisson equation with flower-shaped immersed boundaries
	This test case focuses on second-order immersed boundary methods applied to the Poisson equation with flower-shaped immersed boundaries.
	2D/3D Poisson equation with Robin boundary conditions
	2D/3D Poisson test cases with Robin boundary conditions
	2D-axisymmetric Poisson equation with Dirichlet boundary conditions
	2D-axisymmetric Poisson test cases with Dirichlet boundary conditions
	2D/3D Poisson equation with Neumann boundary conditions
	2D/3D Poisson test cases with Neumann boundary conditions

Detailed Description

Verification of the solution of the Poisson equation (Laplacian operator)

The Poisson equation:

\[ - \Delta T = f \]

is a key equation to solve. It permits us to test the Laplacian operator \( \Delta \equiv \sum_{i=1}^{i=dim} \partial_{x_i x_i} \) schemes involved in various equations (Energy equation, species transport equation, pressure increment or pressure equations) as well as the boundary condition schemes. The equation is solved on various geometries (1D, 2D, 3D and immersed boundaries).

Herein, we propose verification, convergence tests as well as some numerical analysis of the available schemes.