## Example 3

This code solves a system corresponding to a discretization of the Laplace equation with zero boundary conditions on the unit square. The domain is split into an N x N processor grid. Thus, the given number of processors should be a perfect square. Each processor's piece of the grid has n x n cells with n x n nodes connected by the standard 5-point stencil. Note that the struct interface assumes a cell-centered grid, and, therefore, the nodes are not shared. This example demonstrates more features than the previous two struct examples (Example 1 and Example 2). Two solvers are available.

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. We can split the matrix A as

A = [A_ii A_ib; A_bi A_bb].

Let u_0 be the Dirichlet B.C. We can simply say that x_b = u_0. If b_i is the right-hand side, then we just need to solve in the interior:

A_ii x_i = b_i - A_ib u_0.

For this partitcular example, u_0 = 0, so we are just solving A_ii x_i = b_i.

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