Stable and efficient Petrov-Galerkin methods for certain (kinetic) transport equations

Brunken, Julia

Abstract

In this thesis, we develop stable and efficient Petrov-Galerkin discretizations for two different transport-dominated problems: first order linear transport equations and a kinetic Fokker-Planck equation. Based on well-posed weak formulations on the continuous level, the core of our numerical schemes is the choice of the discrete spaces for the Petrov-Galerkin projection. By first defining a discrete test space and then computing a problem-dependent discrete trial space such that the spaces consist of matching stable pairs of trial and test functions, we obtain efficiently computable uniformly stable discrete schemes.For first order linear transport equations, we use an optimally conditioned ultraweak variational formulation. Then, the optimally stable discrete trial space results from the chosen discrete test space by an easy-to-compute application of the adjoint (differential) operator.For the kinetic Fokker-Planck equation, we derive a favorable lower bound for the inf-sup constant on the continuous level with methods inspired by well-posedness results for parabolic equations. Here, the stable discrete trial space is constructed from the test space by the application of the kinetic transport operator and the inverse velocity Laplace-Beltrami operator, so that the specific basis functions can be efficiently computed by low-dimensional elliptic problems.In both cases we thereby guarantee the discrete inf-sup stability with the same inf-sup constant as on the infinite-dimensional level independently of the chosen test spaces. This guaranteed stability is especially beneficial when considering model reduction by the reduced basis method for parametrized first-order transport equations. Using our discretization strategy, we build a reduced model consisting of a fixed reduced test space generated by a greedy algorithm and parameter-dependent reduced trial spaces depending on the test space. Since the stability is inherently built into the method, we can avoid additional stabilization loops within the greedy algorithm, so that we obtain efficient reduced models by an easily implemented procedure.