Continuous interior penalty hp-finite element methods for advection and advection-diffusion equations

A continuous interior penalty $hp$-finite element method that penalizes the jump of the gradient of the discrete solution across mesh interfaces is introduced. Error estimates are obtained for advection and advection--diffusion equations. The analysis relies on three technical results that are of independent interest: an $hp$-inverse trace inequality, a local discontinuous to continuous $hp$-interpolation result, and $hp$-error estimates for continuous $L^2$-orthogonal projections.