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

A continuous interior penalty h p 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 h p hp -inverse trace inequality, a local discontinuous to continuous h p hp -interpolation result, and h p hp -error estimates for continuous L 2 L^2 -orthogonal projections.