Throughout this book, checking convergence numerically has been continually emphasized. However, we have not discussed the more theoretical issues of whether the underlying numerical formulations are indeed convergent, in the sense that the approximate numerical solution fN of the continuous operator equation Lf = g has the property fN → f as N → ∞. The aim of this appendix is to give a brief summary of the current status of this – which readers may be surprised to learn is far from a closed subject.

With the FDTD, the Lax equivalence theorem (discussed in Chapter 2) provides us with confidence that refining the FDTD mesh will indeed result in a convergent solution. With the FEM, work in applied mechanics has provided a rich set of convergence results — although we should note that convergence for high-frequency electromagnetics problems is often in terms of the energy norm, as discussed in Chapter 12. This is a slightly weaker statement of convergence, since the energy norm does not satisfy all the properties of the norm. Also, these proofs are usually in terms of interpolation error; as has been noted, dispersion (or pollution) error is a different problem specific to the differential equation based solvers, but can usually be controlled by adequate meshing.