Hierarchical finite element decomposition of a piecewise smooth function with a discontinuity. The coefficients of the hierarchical finite element approximation of the function become localized as the level increases, i.e., the coefficients corresponding to higher-level basis functions supported away from the discontinuity become small. This makes the detection of discontinuities, an important step in post-processing approximate solutions, a simple matter, unlike what is the case for global spectral basis functions.