Hybridizable discontinuous Galerkin methods for coupled flow and transport systems

Abstract

In this thesis, we propose and analyze hybridizable discontinuous Galerkin methods for coupled flow and transport systems. Such systems may be used to model real-world scenarios in which a fluid contaminant travels through another medium. Common applications include environmental engineering problems and biochemical transport.

This thesis focuses on the Stokes/Darcy-transport and Navier--Stokes/Darcy-transport systems. We consider a two-way coupling between each flow and transport problem: the solution to the flow problem is directly involved in the transport problem, and the solution to the transport problem appears in the flow problem through a parameter function. In each of our considered systems, the flow problem is stationary while the transport problem is time-dependent. The resulting coupled flow and transport systems are quasi-stationary in the sense that the evolution of solutions to the flow problems over time is driven by the transport problem.

Our numerical schemes use a time-lagging method in which the flow and transport problems are decoupled and solved sequentially using hybridizable discontinuous Galerkin methods. This decoupling allows us to establish separate conditions on the discrete flow problem and on the discrete transport problem such that solutions to the combined scheme converge at optimal rates. Moreover, we show how existing results on related discrete flow problems and on the discrete transport problem may be exploited for efficient analysis of the coupled systems. We present this approach in a general setting, and illustrate its use through the specific examples of the Stokes/Darcy-transport and Navier--Stokes/Darcy-transport systems.

For all schemes, we establish the existence of unique numerical solutions over a considered time interval. We prove optimal rates of convergence in space and time, and provide numerical examples to support the theory.