For ray tracing with high accuracy and for the possible application of higher order ray tracing algorithms [1] field calculations are needed with an accuracy of better than 10 to the power of -10. Since field calculations using BEM (boundary element method) can be easily performed for a given number of boundary elements, we have explored the feasibility and the improvement  of accuray by extrapolation to arbitrary fine discretisation. As examples for electrostatic fields we investigated a spherical condensor and specially shaped closed cans, where the analytical fields are known. For magnetic fields we analysed a common magnetic lens with non symmetric pole pieces. By doubling the number of elements, BEM calculations usually improve by more than a factor of 4 in accuracy. However, improvements by orders of magnitude become possible by extrapolating a set of 3 calculations with increasing number of elements. For the magnetic lens example we can demonstrate, how even artifacts, originating from insufficient numerical analysis, can be suppressed by the extrapolation technique.