poisson_1d_multigrid, a Python code which applies a multigrid method to solve the linear system associated with a discretized version of the 1D Poisson equation.
The 1D Poisson equation is assumed to have the form
-u''(x) = f(x), for a < x < b
u(a) = ua, u(b) = ub
Let K be a small positive integer called the mesh index, and let N = 2^K be the corresponding number of uniform subintervals into which [A,B] is divided. Assigning a value U(I) to each of the N+1 equally spaced nodes with coordinate X(I), we approximate the equation by
-U(i-1) + 2 U(i) - U(i+1)
------------------------- = f( X(i) ), 1 < I < N+1
h^2
U(1) = ua, U(N+1) = ub.
It remains to solve the linear system for the desired values of U. This could be done directly, or iteratively. An iterative method such as Jacobi, Gauss-Seidel or SOR might be suitable, but experience shows that the convergence rate of these iterative methods decreases drastically as the value of K is increased - that is, as a more refined and accurate answer is sought.
The multigrid method defines a nested set of grids, and corresponding solutions, to the problem, and applies an iterative linear solver. By transfering information from one grid to a finer or coarser one, a more rapid convergence behavior can be encouraged.
The computer code and data files described and made available on this web page are distributed under the MIT license.
poisson_1d_multigrid is available in a C version and a C++ version and a Fortran77 version and a Fortran90 version and a MATLAB version and an Octave version and a Python version.
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