poisson.hpp File Reference

2D Poisson equation solved via the Discrete Sine Transform. More...

#include "core/matrix.hpp"

Go to the source code of this file.

Namespaces
namespace	num
namespace	num::pde

Functions
Matrix	num::pde::poisson2d_fd (const Matrix &f, int N)
	Solve \(-\Delta u=f\) using finite-difference eigenvalues.
Matrix	num::pde::poisson2d (const Matrix &f, int N)
	Solve \(-\Delta u=f\) using continuous eigenvalues \((k\pi)^2\).

Detailed Description

2D Poisson equation solved via the Discrete Sine Transform.

Solves the homogeneous Dirichlet problem

\[ -\Delta u(x,y) = f(x,y), \qquad (x,y) \in (0,1)^2, \qquad u|_{\partial\Omega}=0 . \]

On an \(N \times N\) interior grid with \(h = 1/(N+1)\), the finite difference variant diagonalizes

\[ L_{2D} = L_{1D} \otimes I + I \otimes L_{1D} \]

with the DST-I basis

\[ \phi_k(j) = \sin\!\left(\frac{j k \pi}{N+1}\right). \]

The transformed solve is

\[ \hat{u}_{pq} = \frac{h^2 \hat{f}_{pq}} {2(1-\cos(p\pi/(N+1))) + 2(1-\cos(q\pi/(N+1)))} . \]

Definition in file poisson.hpp.