Stabilized finite element methods for nonsymmetric, noncoercive and ill-posed problem. Part II: hyperbolic equations

TL;DR

This paper analyzes stabilized finite element methods for hyperbolic transport equations lacking coercivity, establishing convergence conditions and demonstrating their applicability to various stabilized methods through theoretical and numerical validation.

Contribution

It introduces abstract convergence conditions for stabilized finite element methods applied to noncoercive hyperbolic equations, including new optimization-based stabilization techniques.

Findings

Convergence conditions are verified for three stabilized methods.

The optimization-based stabilization method is effective for hyperbolic equations.

Numerical examples confirm theoretical predictions.

Abstract

In this paper we consider stabilised finite element methods for hyperbolic transport equations without coercivity. Abstract conditions for the convergence of the methods are introduced and these conditions are shown to hold for three different stabilised methods: the Galerkin least squares method, the continuous interior penalty method and the discontinuous Galerkin method. We consider both the standard stabilisation methods and the optimisation based method introduced in \cite{part1}. The main idea of the latter is to write the stabilised method in an optimisation framework and select the discrete function for which a certain cost functional, in our case stabilisation term, is minimised. Some numerical examples illustrate the theoretical investigations.