  Collocation boundary element methods for integral equations are easier to
implement than Galerkin methods because the elements of the discretization
matrix are given by lower-dimensional integrals. For that same reason, the
matrix assembly also requires fewer computations. However, collocation methods
typically yield slower convergence rates and less robustness, compared to
Galerkin methods. We explore the extent to which oversampled collocation can
improve both robustness and convergence rates. We show that in some cases
convergence rates can actually be higher than the corresponding Galerkin
method, although this requires oversampling at a faster than linear rate. In
most cases of practical interest, oversampling at least lowers the error by a
constant factor. This can still be a substantial improvement: we analyze an
example where linear oversampling by a constant factor $J$ (leading to a
rectangular system of size $JN \times N$) improves the error at a cubic rate in
the constant $J$. Furthermore, the oversampled collocation method is much less
affected by a poor choice of collocation points, as we show how oversampling
can lead to guaranteed convergence. Numerical experiments are included for the
two-dimensional Helmholtz equation.
