Supercloseness of the continuous interior penalty method for singularly perturbed problems in 1D: Vertex-cell interpolation

A continuous interior penalty method with piecewise polynomials of degree p≥2 is applied on a Shishkin mesh to solve a singularly perturbed convection-diffusion problem, whose solution has a single boundary layer. This method is analyzed by means of a series of integral identities developed for the convection terms. Then we prove a supercloseness bound of order 5/2 for a vertex-cell interpolation when p=2. The sharpness of our analysis is supported by some numerical experiments. Moreover, numerical tests show supercloseness clearly for p≥3.