(Parametrized) First Order Transport Equations: Realization of Optimally Stable Petrov-Galerkin Methods

TL;DR

This paper introduces a new class of ultraweak variational formulations for parametrized first order transport equations, achieving optimal stability and simplifying computational realization, especially in reduced basis methods.

Contribution

It develops computationally feasible, optimally stable Petrov-Galerkin methods with inf-sup constant one, avoiding stabilization loops in reduced basis model construction.

Findings

Inf-sup constant is one in continuous and discrete cases

Method demonstrates good performance in numerical experiments

Avoids stabilization loop in reduced basis greedy algorithms

Abstract

We consider ultraweak variational formulations for (parametrized) linear first order transport equations in time and/or space. Computationally feasible pairs of optimally stable trial and test spaces are presented, starting with a suitable test space and defining an optimal trial space by the application of the adjoint operator. As a result, the inf-sup constant is one in the continuous as well as in the discrete case and the computational realization is therefore easy. In particular, regarding the latter, we avoid a stabilization loop within the greedy algorithm when constructing reduced models within the framework of reduced basis methods. Several numerical experiments demonstrate the good performance of the new method.