This provides two multigrid solvers for cell-centered data.

MG.py

This is the basic solver. It solves constant coefficient problems of the form:

(alpha - beta L) phi = f

where L is the Laplacian.

The following drivers test it:

• mg_test_simple.py: this solves:

  u_xx + u_yy = -2[(1-6x**2)y**2(1-y**2) + (1-6y**2)x**2(1-x**2)]

  at a single resolution on [0,1]x[0,1], with u = 0 on the boundary (Dirichlet BCs).

• mg_vis.py: this solves the same problem as mg_test_simple.py, but it outputs a detailed set of plots at each smoothing iteration showing the progression of the solve through the V-cycles

variable_coeff_MG.py

This is a variable-coefficient solver. It subclasses MG.py and extends the basic solver to solve problems with variable coefficients:

D (eta G phi) = f

where D is the divergence and G is the gradient.

The following drivers test it:

• mg_test_vc_constant.py: this solves the same constant-coefficnet Poisson problem as mg_test_simple.py, but using the variable_coeff_MG.py framework. This makes sure that we can fall back to the simpler constant-coefficient case.

• mg_test_vc_dirichlet.py: this solves

  div . ( alpha grad phi ) = f

  with

  alpha = 2.0 + cos(2.0*pi*x)*cos(2.0*pi*y)
  f = -16.0*pi**2*(cos(2*pi*x)*cos(2*pi*y) + 1)*sin(2*pi*x)*sin(2*pi*y)
  

  on [0,1] x [0,1] with homogeneous Dirichlet BCs. The solution is compared to the exact solution and the convergence is measured.

• mg_test_vc_periodic.py: this solves the same problem as mg_test_vc_dirichlet.py, but with periodic boundary conditions.

general_MG.py

This is the general multigrid solver, designed to solve elliptic problems of the form:

alpha phi + div . (beta grad phi) + gamma . grad phi = f

The following drivers test it:

• mg_test_general_alphabeta_only.py: this neglects the gamma term and solves:

  alpha phi + div . ( beta grad phi ) = f

  with

  alpha = 1.0
  beta = 2.0 + cos(2.0*pi*x)*cos(2.0*pi*y)
  f = (-16.0*pi**2*cos(2*pi*x)*cos(2*pi*y) - 16.0*pi**2 + 1.0)*sin(2*pi*x)*sin(2*pi*y)
  

  on [0,1] x [0,1] with homogeneous Dirichlet BCs. The solution is compared to the exact solution and the convergence is measured.

• mg_test_general_beta_only.py: this neglects both the alpha and gamma terms and solves the same variable-coefficient Poisson problem as mg_test_vc_dirichlet.py (note that the naming of the coefficients differ between those solvers, but the equation solved is the same).

• mg_test_general_constant.py: this solves a pure Poisson problem (alpha = gamma = 0; beta = 1), solving the same problem as the base MG solver in mg_test_simple.py.

• mg_test_general_dirichlet.py: This solves a general elliptic problem of the form:

  alpha phi + div{beta grad phi} + gamma . grad phi = f

  with

  alpha = 1.0
  beta = cos(2*pi*x)*cos(2*pi*y) + 2.0
  gamma_x = sin(2*pi*x)
  gamma_y = sin(2*pi*y)
  
  f = (-16.0*pi**2*cos(2*pi*x)*cos(2*pi*y) + 2.0*pi*cos(2*pi*x) +
        2.0*pi*cos(2*pi*y) - 16.0*pi**2 + 1.0)*sin(2*pi*x)*sin(2*pi*y)
  

  on [0,1] x [0,1] with homogeneous Dirichlet BCs. The solution is compared to the exact solution and a convergence plot is made.

• mg_test_general_inhomogeneous.py: This solves a general elliptic problem with inhomogeneous BCs. The coefficients are:

  alpha = 10.0
  beta = x*y + 1  (note: x*y alone doesn't work)
  gamma_x = 1
  gamma_y = 1
  
  f =  -(pi/2)*(x + 1)*sin(pi*y/2)*cos(pi*x/2)
      - (pi/2)*(y + 1)*sin(pi*x/2)*cos(pi*y/2) +
      (-pi**2*(x*y+1)/2 + 10)*cos(pi*x/2)*cos(pi*y/2)
  

  on [0,1] x [0,1], with Dirichlet BCs of the form:

  phi(x=0) = cos(pi*y/2)
  phi(x=1) = 0
  phi(y=0) = cos(pi*x/2)
  phi(y=1) = 0
  

prolong_restrict_demo.py

This tests that the restriction and prolongation operations work as expected.