# poisson_1d_multigrid

poisson_1d_multigrid, a FORTRAN90 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.

### Languages:

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.

### Related Data and Programs:

cyclic_reduction, a FORTRAN90 code which solves a tridiagonal linear system using cyclic reduction.

FD1D_BVP, a FORTRAN90 code which applies the finite difference method to a two point boundary value problem in one spatial dimension.

MGMRES, a FORTRAN90 code which applies the restarted GMRES algorithm to solve a sparse linear system, by Lili Ju.

### Reference:

1. William Briggs, Van Emden Henson, Steve McCormick,
A Multigrid Tutorial,
SIAM, 2000,
ISBN13: 978-0-898714-62-3,
LC: QA377.B75.
2. William Hager,
Applied Numerical Linear Algebra,
Prentice-Hall, 1988,
ISBN13: 978-0130412942,
LC: QA184.H33.

### Source Code:

Last revised on 05 August 2022.