The Multilevel Multigraph Iterative Method


In this example, we show the convergence history of the multilevel-multigraph iterative method on a selection of simple partial differential equations. All are solved on the this 101 x 101 uniform triangulation of the unit square. All problems have homogeneous Dirichlet boundary conditions. The multilevel multigraph iterative method based on principles of classical multigrid methods. However, it is completely algebraic in nature. The smoother is an ILU using minimum degree ordering and a drop tolerance (in this case 1.e-2). The coarsening procedure is based on the ILU factorization in order to reduce the setup overhead.

conv1 soln1

This problem was -Laplace u = 1.

conv2 soln2

This problem was -Laplace u +1000 u_x = 1.

conv3 soln3

This problem was -Laplace u + 1000 u_x + 1000 u_y =1.

conv4 soln4

This problem was -Laplace u - 1000 u = 1.

conv5 soln5

This problem was -Laplace u + 1000 u = 1.

conv6 soln6

This problem was -.001 u_xx - u_yy = 1.

conv7 soln7

This problem was -Laplace u + 1000 ( u_x (y-.5) - u_y (x-.5) ) = 1.