The Convection-Diffusion Equation

A simple modification of the Poisson equation, namely the addition of a first derivative term in the p.d.e., leads to a non-symmetric matrix, and can be used to test the non-symmetric solver capabilities of the software. The parameter $\alpha$ controls the degree of non-symmetry in this test problem.

$$\begin{eqnarray*} -\nabla^2 u(x) + \alpha \left(\frac{\partial u(x)}{\partial... ...}{\partial z} \right) &=& f(x), \quad \mbox{in} \quad \Omega. \end{eqnarray*}$$

The sparsity structure of resulting matrix is the same as in the Poisson problem. In particular, we can use either the AIJ format or the structured grid features of PETSc and Hypre.