(Parametrized) First Order Transport Equations: Realization of Optimally Stable Petrov-Galerkin Methods
For researchers working on reduced basis methods for transport equations, this work offers a practical way to achieve optimal stability without iterative stabilization.
The paper presents computationally feasible pairs of optimally stable trial and test spaces for ultraweak variational formulations of (parametrized) linear first order transport equations, achieving an inf-sup constant of one in both continuous and discrete cases, which simplifies computational realization and avoids stabilization loops in reduced basis methods.
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.