  We present and analyse a numerical framework for the approximation of
nonlinear degenerate elliptic equations of the Stefan or porous medium types.
This framework is based on piecewise constant approximations for the functions,
which we show are essentially necessary to obtain convergence and error
estimates. Convergence is established without regularity assumption on the
solution. A detailed analysis is then performed to understand the design
properties that enable a scheme, despite these piecewise constant
approximations and the degeneracy of the model, to satisfy high-order error
estimates if the solution is piecewise smooth. Numerical tests, based on
continuous and discontinuous approximation methods, are provided on a variety
of 1D and 2D problems, showing the influence on the convergence rate of the
nature of the degeneracy and of the design choices.
