|Title:|| Interior Penalty Continuous and Discontinuous Finite Element Approximations of Hyperbolic Equations|
|Date:|| Wednesday 31st October 2007|
|Author(s) :|| Burman, Erik; Quarteroni, Alfio; Stamm, Benjamin|
|Abstract:|| In this paper we present the continuous and discontinuous Galerkin
methods in a unified setting for the numerical approximation of the transport
dominated advection-reaction equation. Both methods are stabilized by the
interior penalty method, more precisely by the jump of the gradient in the continuous
case whereas in the discontinuous case the stabilization of the jump of
the solution and optionally of its gradient is required to achieve optimal convergence.
We prove that the solution in the case of the continuous Galerkin
approach can be considered as a limit of the discontinuous one when the stabilization
parameter associated with the penalization of the solution jump tends
to infinity. As a consequence, the limit of the numerical flux of the discontinuous
method yields a numerical flux for the continuous method too. Numerical
results will highlight the theoretical results that are proven in this paper.