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.
- Linear transport equation
- Inf-sup stability
- Reduced basis methods
Brunken, J., Smetana, K., & Urban, K. (2019). (Parametrized) First Order Transport Equations: Realization of Optimally Stable Petrov-Galerkin Methods. SIAM journal on scientific computing, 41(1), A592-A621. https://doi.org/10.1137/18M1176269