The Material Point Method (MPM) is widely used for challenging applications in engineering, and animation but lags behind some other methods in terms of error analysis and computable error estimates. The complexity and nonlinearity of the equations solved by the method and its reliance both on a mesh and on moving particles makes error estimation challenging. Error analysis of a simple MPM method has shown the global error to have variable order between frist and second error fro widley used methods. The overall time dependent nature of MPM also complicates matters as both space and time errors and their evolution must be considered thus leading to the use of explicit error transport equations. The preliminary use of an error estimator based on this transport approach has yielded promising results in the 1D case including cases where the error deteriorate due to grid-crossing error. the extension of the error estimation approach to a well-known model problem in two space higher dimensions is considered and together with additional algorithmic and theoretical results, shown to give promising results in preliminary computational experiments.