Weighted error estimates for transient transport problems discretized using continuous finite elements with interior penalty stabilization on the gradient jumps

Abstract

In this paper we consider the semi-discretization in space of a first order scalar transport equation. For the space discretization we use standard continuous finite elements. To obtain stability we add a penalty on the jump of the gradient over element faces. We recall some global error estimates for smooth and rough solutions and then prove a new local error estimate for the transient linear transport equation. In particular we show that in the stabilized method the effect of non-smooth features in the solution decay exponentially from the space time zone where the solution is rough so that smooth features will be transported unperturbed. Locally the L 2 -norm error converges with the expected order O(h k+ 1 2 ). We then illustrate the results numerically. In particular we show the good local accuracy in the smooth zone of the stabilized method and that the standard Galerkin fails to approximate a solution that is smooth at the final time if discontinuities have been present in the solution at some time during the evolution.

Spectral analysis of high order continuous FEM for hyperbolic PDEs on triangular meshes: influence of approximation, stabilization, and time-stepping

“…In this direction, it would be interesting to evaluate the stability of the CIP adding a additional penalty term on the jump of higher order derivatives as suggested in [17,13,3]. Moreover, it could be interesting to see the stability of Cubature elements using higher degree polynomials.…”

“…Note that for higher than second order it may be relevant to consider additional penalty terms based on higher derivatives (see e.g. [17,13,3]). We did not do this in this work.…”

“…This type of analysis has been already done for multidimensional problems inter alia in [2]. More rigorous proof of error bounds with h p+ 1 2 estimates can be found in [13] for the CIP. We did not consider in this work projection stabilizations involving higher derivatives.…”