The dynamics of many physical problems are modelled by partial differential equations. These models have an infinite-dimensional state-space. In general, it is necessary to use a numerical approximation to simulate the response of the system and to compute controllers for the system. However, a scheme that yields good results when used for simulation may be inappropriate for use in controller design. For instance, a scheme for which the simulation results converge quickly may have a corresponding controller sequence that converges very slowly or not at all. Criteria appropriate for evaluation of an approximation scheme will be discussed. These ideas will be applied to the problem of designing a controller to solve a $\mathcal{H}^\infty$ disturbance-attenuation problem.