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 on (0,1)^2 (Dirichlet) via DST with FD eigenvalues.
Matrix	num::pde::poisson2d (const Matrix &f, int N)
	Solve -Delta u = f on (0,1)^2 (Dirichlet) via DST with exact eigenvalues.

Detailed Description

2D Poisson equation solved via the Discrete Sine Transform.

Solves -Delta u = f on (0,1)^2 with homogeneous Dirichlet BC, discretised on an N x N interior grid with h = 1/(N+1). N must equal 2^p - 1 so that the odd-extension length 2(N+1) is a power of two (required by the radix-2 FFT backend).

Algorithm (Demmel, CS267 Lecture 20): L_2 = L_1 x I + I x L_1, L_1 = tridiag(-1, 2, -1). L_1 = F * D * F^T, F_{jk} = sin(j*k*pi/(N+1)). Transformed system: u_hat_{jk} = h^2 * f_hat_{jk} / (D_j + D_k). DST-I computed via complex FFT on an odd-extended sequence. Cost: O(N^2 log N).

Two variants: poisson2d_fd – FD eigenvalues D_k = 2(1 - cos(k*pi/(N+1))). Error O(h^2). poisson2d – Exact eigenvalues (k*pi)^2. Machine-precision error for f in the DST eigenbasis; exponential convergence otherwise.

Reference: J. Demmel, CS267 Lecture 20, UC Berkeley, 2025.

Definition in file poisson.hpp.