A Hybridized Discontinuous Galerkin Scheme for the Coupled Stokes-Darcy Flow and Transport

The main focus of this thesis is on finding highly accurate and robust numerical methods to solve a complex flow and transport problem governed by the fully-coupled time-dependent Stokes-Darcy-transport equations. This problem has many applications one of which is groundwater contamination by pollutants transported via surface/subsurface flow. It consists of two main ingredients; the time-dependent Stokes-Darcy equations describing the flow, and the time-dependent advection-diffusion equation for the transport of chemicals via this flow. Therefore, the first part of this thesis is dedicated to studying the time-dependent Stokes-Darcy problem that describes the free flow and porous media flow on two different parts of a domain and their interaction at the common interface. We introduce a hybridized discontinuous Galerkin (HDG) method which provides exact mass conservation and pressure robustness and handles the interface conditions via facet unknowns. We prove well-posedness and a priori error estimates in the energy norm, and provide numerical experiments that show optimal convergence and robustness of the method with respect to the problem parameters. The second part deals with the time-dependent advection-diffusion equation where we again use an HDG method for the spatial discretization. We show the existence and uniqueness of the semi-discrete transport problem and prove a priori error estimates in the energy norm. A number of numerical experiments are presented for different boundary conditions and we observe optimal rates of convergence in each case. Combining the two parts by a sequential algorithm, we solve the fully coupled time-dependent Stokes-Darcy-transport problem. The coupling of the flow and transport is introduced by the dependence of the fluid viscosity and source/sink terms on the concentration and by the dependence of the dispersion/diffusion tensor in the porous media domain on the advective fluid velocity. Our sequential algorithm employs a linearizing decoupling strategy based on the backward Euler time-stepping where the Stokes-Darcy and the transport equations are solved sequentially by time-lagging the concentration. The well-posedness results and a priori error estimates for the velocity and the concentration in the energy norm are presented and numerical examples demonstrating optimal convergence and mass conservation are provided.