Simple Multigrid Examples#

Known Solution#

A basic multigrid test is run as (using a path relative to the root of the pyro2 repository):

./pyro/multigrid/examples/mg_test_simple.py

The mg_test_simple.py script solves a Poisson equation with a known analytic solution. This particular example comes from the text A Multigrid Tutorial, 2nd Ed., by Briggs. The example is:

u_{x x} + u_{y y} = - 2 [(1 - 6 x^{2}) y^{2} (1 - y^{2}) + (1 - 6 y^{2}) x^{2} (1 - x^{2})]

on $[0, 1] \times [0, 1]$ with $u = 0$ on the boundary.

The solution to this is shown below.

Since this has a known analytic solution:

u (x, y) = (x^{2} - x^{4}) (y^{4} - y^{2})

We can assess the convergence of our solver by running at a variety of resolutions and computing the norm of the error with respect to the analytic solution. This is shown below:

The dotted line is 2nd order convergence, which we match perfectly.

The movie below shows the smoothing at each level to realize this solution:

You can run this example locally by running the mg_vis.py script:

./pyro/multigrid/examples/mg_vis.py

Projection#

Another example uses multigrid to extract the divergence free part of a velocity field. The script to run this is project_periodic.py. This is run as:

./pyro/multigrid/examples/project_periodic.py

Given a vector field, $U$ , we can decompose it into a divergence free part, $U_{d}$ , and the gradient of a scalar, $ϕ$ :

U = U_{d} + \nabla ϕ

We can project out the divergence free part by taking the divergence, leading to an elliptic equation:

\nabla^{2} ϕ = \nabla \cdot U

The project-periodic.py script starts with a divergence free velocity field, adds to it the gradient of a scalar, and then projects it to recover the divergence free part. The error can found by comparing the original velocity field to the recovered field. The results are shown below:

Left is the original u velocity, middle is the modified field after adding the gradient of the scalar, and right is the recovered field.

Exercises#

Explorations#

Try doing just smoothing, no multigrid. Show that it still converges second order if you use enough iterations, but that the amount of time needed to get a solution is much greater.

Extensions#

Add a different bottom solver to the multigrid algorithm
Make the multigrid solver work for non-square domains
Implement the full-multigrid algorithm instead of just V-cycles

Simple Multigrid Examples

Contents