# Multigrid solvers¶

pyro solves elliptic problems (like Laplace’s equation or Poisson’s equation) through multigrid. This accelerates the convergence of simple relaxation by moving the solution down and up through a series of grids. Chapter 9 of the pdf notes gives an introduction to solving elliptic equations, including multigrid.

There are three solvers:

The core solver, provided in the class

`MG.CellCenterMG2d`

solves constant-coefficient Helmholtz problems of the form \((\alpha - \beta \nabla^2) \phi = f\)The class

`variable_coeff_MG.VarCoeffCCMG2d`

solves variable coefficient Poisson problems of the form \(\nabla \cdot (\eta \nabla \phi ) = f\). This class inherits the core functionality from`MG.CellCenterMG2d`

.The class

`general_MG.GeneralMG2d`

solves a general elliptic equation of the form \(\alpha \phi + \nabla \cdot ( \beta \nabla \phi) + \gamma \cdot \nabla \phi = f\). This class inherits the core functionality from`MG.CellCenterMG2d`

.This solver is the only one to support inhomogeneous boundary conditions.

We simply use V-cycles in our implementation, and restrict ourselves to square grids with zoning a power of 2.

The multigrid solver is not controlled through pyro.py since there is no time-dependence in pure elliptic problems. Instead, there are a few scripts in the multigrid/ subdirectory that demonstrate its use.

## Examples¶

### multigrid test¶

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

repository):

```
./examples/multigrid/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:

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:

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:

```
./examples/multigrid/mg_vis.py
```

### projection¶

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

```
./examples/multigrid/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, \(\phi\):

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

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¶

Implement inhomogeneous dirichlet boundary conditions

Add a different bottom solver to the multigrid algorithm

Make the multigrid solver work for non-square domains