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

TL;DR

This paper develops and analyzes a stabilized finite element method for scalar transport equations, demonstrating exponential decay of non-smooth features and optimal local error convergence, with numerical validation.

Contribution

It introduces a new local error estimate for stabilized finite element discretizations of transient transport problems, highlighting the decay of non-smooth features and improved accuracy.

Findings

Exponential decay of non-smooth features in stabilized method

Local error convergence of order O(h^{k+1/2})

Standard Galerkin fails for solutions with past discontinuities

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 $O(h^{k+\frac12})$. 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.