A priori error estimates for interior penalty versions of the local discontinuous Galerkin method applied to transport equations (Q2762696)

A special version of a local discontinuous Galerkin method for convection-diffusion equations is investigated. This method was developed in a work of \textit{B. Cockburn} and \textit{C.-W. Shu} [SIAM J. Numer. Anal. 35, No. 6, 2440-2463 (1998; Zbl 0927.65118)]. In this paper a version with interior penalties for the numerical solution of transport equations is considered. Two interior penalty methods are depicted: one, that penalizes jumps in the solution accross interelement boundaries, and another that also penalizes jumps in the diffusive flux across such boundaries. A priori error estimates are proved for both penalty methods: for the first penalty method convergence of order \(k\) in \( L^{\infty}(L^2)\) norm for polynomials of minimal degree \(k\) are used. For the second penalty method convergence of order \(k+1/2 \) in norm \(L^2(L^2)\) is proved.