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

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:

./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.. The example is:

\[u_{xx} + u_{yy} = -2 \left [(1-6x^2)y^2(1-y^2) + (1-6y^2)x^2(1-x^2)\right ]\]

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

The solution to this is shown below.

_images/mg_test.png

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:

_images/mg_converge.png

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

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


projection

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

./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, \(\phi\):

\[U = U_d + \nabla \phi\]

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

\[\nabla^2 \phi = \nabla \cdot U\]

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.

Jupyter Notebook

A jupyter notebook showing how to use the basic solver can be found here: multigrid-examples.ipynb

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