The Poisson Model Problem

The symmetric linear system is derived by a standard 7-point discretization of the Poisson equation, with homogeneous Dirichlet boundary conditions, using a uniform grid[8].

$$\begin{eqnarray*}-\nabla^2 u(x) = -\frac{\partial^2 u(x)}{\partial x^2} -\frac{\... ...a = (0,1)^3 \\ u(x) &=& 0 \quad \mbox{on} \quad \partial \Omega. \end{eqnarray*}$$

The total number of unknowns is $N = nx \times ny \times nz$.

The sparse matrix resulting from this discretization can be stored in a standard compressed row sparse storage format (the AIJ format), or alternatively, by using the structured grid infrastructure in PETSc or Hypre.