Space-time Hybridizable Discontinuous Galerkin Method for the Advection-Diffusion Problem

Abstract

In this thesis, we analyze a space-time hybridizable discontinuous Galerkin (HDG) method for the time-dependent advection-dominated advection-diffusion problem. It is well-known that solutions to these problems may admit sharp boundary and interior layers and that many numerical methods are prone to non-physical oscillations when resolving these solutions. This challenge has prompted the design of many new numerical methods and stabilization mechanisms. Among others, HDG methods prove to be capable of resolving the sharp layers in a robust manner. The design principles of HDG methods consist of discontinuous Galerkin (DG) methods and their strong stability properties, as well as hybridization to reduce the computational cost of the numerical method.

The analysis in this work focuses on a space-time formulation of the time-dependent advection-diffusion problem and an HDG discretization in both space and time. This provides a straightforward approach to discretize the problem on a time-dependent domain, with arbitrary higher-order spatial and temporal accuracy. We present an a priori error analysis that provides Peclet-robust error estimates that are also valid on moving meshes. A key intermediate step towards our error estimates is a Peclet-robust inf-sup stability condition.

The second contribution of this thesis is an a posteriori error analysis of the space-time HDG method for the time-dependent advection-dominated advection-diffusion problem on fixed domains. This is motivated by the efficiency of combining a posteriori error estimators with adaptive mesh refinement (AMR) to locally refine or coarsen a mesh in the presence of sharp layers. When the solution admits sharp layers, AMR may still lead to optimal rates of convergence in terms of the number of degrees-of-freedom, unlike uniform mesh refinement.

In this thesis, we present an a posteriori error estimator for the space-time HDG method with respect to a locally computable norm. We prove its reliability and local efficiency. The proof of reliability is based on a combination of a Peclet-robust coercivity type result and a saturation assumption. In addition, efficiency, which is local both in space and time, is shown using bubble function techniques. The error estimator in this thesis is fully local, hence it is an estimator for local space and time adaptivity in the AMR procedure.

Finally, numerical simulations are presented to demonstrate and verify the theory. Both uniform and adaptive refinement strategies are performed on problems which admit boundary and interior layers.