Steps to solve a PDE

1. Discretization of the PDE

• Analytical Problem

$$\Delta u=f$$

Note: $f$ is the source term.

• Discrete Problem

$$\mathbf{Au}=\mathbf{f}$$

Note: $\mathbf{A}$ is the finite dimensional representation of $\Delta$ and includes the boundary condition.

2. Find Approximate solution

• A problem suited algorithm needs to be chosen to compute.

$$\mathbf{u}={\mathbf{A}}^{-1}\mathbf{f}$$

3. Evaluate the solution $u$

• $\mathbf{u}$ is plotted.