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

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AbstractThe local discontinuous Galerkin method has been developed recently by Cockburn and Shu for convection‐dominated convection‐diffusion equations. In this article, we consider versions of this method with interior penalties for the numerical solution of transport equations, and derive a priori error estimates. We consider two interior penalty methods, one that penalizes jumps in the solution across interelement boundaries, and another that also penalizes jumps in the diffusive flux across such boundaries. For the first penalty method, we demonstrate convergence of order k in the L∞(L2) norm when polynomials of minimal degree k are used, and for the second penalty method, we demonstrate convergence of order k+1/2. Through a parabolic lift argument, we show improved convergence of order k+1/2 (k+1) in the L2(L2) norm for the first penalty method with a penalty parameter of order one (h−1). © 2001 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 17: 545–564, 2001