**poisson_2d**,
a Fortran90 code which
computes an approximate solution to the Poisson equation
in a rectangular region.

The version of Poisson's equation being solved here is

- ( d/dx d/dx + d/dy d/dy ) U(x,y) = F(x,y)over the rectangle 0 <= X <= 1, 0 <= Y <= 1, with exact solution

U(x,y) = sin ( pi * x * y )so that

F(x,y) = pi^2 * ( x^2 + y^2 ) * sin ( pi * x * y )and with Dirichlet boundary conditions along the lines x = 0, x = 1, y = 0 and y = 1. (The boundary conditions will actually be zero in this case, but we write up the problem as though we didn't know that, which makes it easy to change the problem later.)

We compute an approximate solution by discretizing the geometry, assuming that DX = DY, and approximating the Poisson operator by

( U(i-1,j) + U(i+1,j) + U(i,j-1) + U(i,j+1) - 4*U(i,j) ) / dx /dyAlong with the boundary conditions at the boundary nodes, we have a linear system for U. We can apply the Jacobi iteration to estimate the solution to the linear system.

The information on this web page is distributed under the MIT license.

**poisson_2d** is available in
a C version and
a Fortran90 version and
a FENICS version and
a FreeFem version and
a MATLAB version and
an Octave version and
a Python version.

poisson_1d, a Fortran90 code which solves a discretized version of the Poisson equation -uxx = f(x) on the interval a ≤ x ≤ b, with Dirichlet boundary conditions u(a) = ua, u(b) = ub. The linear system is solved using Gauss-Seidel iteration.

poisson_openmp, a Fortran77 code which computes an approximate solution to the Poisson equation in a rectangle, using the Jacobi iteration to solve the linear system, and OpenMP to carry out the Jacobi iteration in parallel.

- poisson_2d.f90, the source code.
- poisson_2d.sh, compiles the source code.