  In this paper, we want to clarify the Gibbs phenomenon when continuous and
discontinuous finite elements are used to approximate discontinuous or nearly
discontinuous PDE solutions from the approximation point of view.
  For a simple step function, we explicitly compute its continuous and
discontinuous piecewise constant or linear projections on discontinuity matched
or non-matched meshes. For the simple discontinuity-aligned mesh case,
piecewise discontinuous approximations are always good. For the general
non-matched case, we explain that the piecewise discontinuous constant
approximation combined with adaptive mesh refinements is a good choice to
achieve accuracy without overshoots. For discontinuous piecewise linear
approximations, non-trivial overshoots will be observed unless the mesh is
matched with discontinuity. For continuous piecewise linear approximations, the
computation is based on a "far-away assumption", and non-trivial overshoots
will always be observed under regular meshes. We calculate the explicit
overshoot values for several typical cases. Numerical tests are conducted for a
singularly-perturbed reaction-diffusion equation and linear hyperbolic
equations to verify our findings in the paper. Also, we discuss the
$L^1$-minimization-based methods and do not recommend such methods due to their
similar behavior to $L^2$-based methods and more complicated implementations.
