  We study the strong approximation of stochastic differential equations with
discontinuous drift coefficients and (possibly) degenerate diffusion
coefficients. To account for the discontinuity of the drift coefficient we
construct an adaptive step sizing strategy for the explicit Euler-Maruyama
scheme. As a result, we obtain a numerical method which has -- up to
logarithmic terms -- strong convergence order $1/2$ with respect to the average
computational cost. We support our theoretical findings with several numerical
examples.
