Discontinuous Galerkin error estimation for linear symmetrizable hyperbolic systems

Abstract:

We present an a posteriori error analysis for the discontinuous Galerkin discretization error of first-order linear symmetrizable hyperbolic systems of partial differential equations with smooth solutions. We perform a local error analysis by writing the local error as a series and showing that its leading term can be expressed as a linear combination of Legendre polynomials of degree $p$ and $p+1$. We apply these asymptotic results to show that projections of the error are pointwise $\mathcal {O}(h^{p+2})$-superconvergent. We solve relatively small local problems to compute efficient and asymptotically exact estimates of the finite element error. We present computational results for several linear hyperbolic systems in acoustics and electromagnetism.