CG Example Codes

Example 1

This is a two processor example. Each processor owns one box in the grid. For reference, the two grid boxes are those in the example diagram in the struct interface chapter of the User's Manual. Note that in this example code, we have used the two boxes shown in the diagram as belonging to processor 0 (and given one box to each processor). The solver is PCG with no preconditioner.

We recommend viewing examples 1-4 sequentially for a nice overview/tutorial of the struct interface.

Example 4

This example differs from the previous structured example (Example 3) in that a more sophisticated stencil and boundary conditions are implemented. The method illustrated here to implement the boundary conditions is much more general than that in the previous example. Also symmetric storage is utilized when applicable.

This code solves the convection-reaction-diffusion problem div (-K grad u + B u) + C u = F in the unit square with boundary condition u = U0. The domain is split into N x N processor grid. Thus, the given number of processors should be a perfect square. Each processor has a n x n grid, with nodes connected by a 5-point stencil. Note that the struct interface assumes a cell-centered grid, and, therefore, the nodes are not shared.

To incorporate the boundary conditions, we do the following: Let x_i and x_b be the interior and boundary parts of the solution vector x. If we split the matrix A as

A = [A_ii A_ib; A_bi A_bb],

then we solve

[A_ii 0; 0 I] [x_i ; x_b] = [b_i - A_ib u_0; u_0].

Note that this differs from the previous example in that we are actually solving for the boundary conditions (so they may not be exact as in ex3, where we only solved for the interior). This approach is useful for more general types of b.c.

A number of solvers are available. More information can be found in the Solvers and Preconditioners chapter of the User's Manual.

We recommend viewing examples 1, 2, and 3 before viewing this example.

Example 5

This example solves the 2-D Laplacian problem with zero boundary conditions on an nxn grid. The number of unknowns is N=n^2. The standard 5-point stencil is used, and we solve for the interior nodes only.

This example solves the same problem as Example 3. Available solvers are AMG, PCG, and PCG with AMG or Parasails preconditioners.

Example 7

This example uses the sstruct interface to solve the same problem as was solved in Example 4 with the struct interface. Therefore, there is only one part and one variable.

This code solves the convection-reaction-diffusion problem div (-K grad u + B u) + C u = F in the unit square with boundary condition u = U0. The domain is split into N x N processor grid. Thus, the given number of processors should be a perfect square. Each processor has a n x n grid, with nodes connected by a 5-point stencil. We use cell-centered variables, and, therefore, the nodes are not shared.

To incorporate the boundary conditions, we do the following: Let x_i and x_b be the interior and boundary parts of the solution vector x. If we split the matrix A as

A = [A_ii A_ib; A_bi A_bb],

then we solve

[A_ii 0; 0 I] [x_i ; x_b] = [b_i - A_ib u_0; u_0].

Note that this differs from the previous example in that we are actually solving for the boundary conditions (so they may not be exact as in ex3, where we only solved for the interior). This approach is useful for more general types of b.c.

As in the previous example (Example 6), we use a structured solver. A number of structured solvers are available. More information can be found in the Solvers and Preconditioners chapter of the User's Manual.

We recommend viewing Examples 6 before viewing this example.