Research works on the coupling of incompressible surface flow with subsurface porous media flow arouse increasing interest recently. The coupled problem is a typical multi-domain problem with multi-physics. In-depth understanding of this problem requires both modeling process and numerical study. In this work, some existing surface flow models and subsurface flow models are reviewed; the interaction mechanisms of surface flow with subsurface porous media flow are discussed; numerical algorithms for solving coupled surface/subsurface flow models are proposed; in particular, preconditioning techniques and two grid algorithms are mathematically and numerically investigated....[ Read more ]

Research works on the coupling of incompressible surface flow with subsurface porous media flow arouse increasing interest recently. The coupled problem is a typical multi-domain problem with multi-physics. In-depth understanding of this problem requires both modeling process and numerical study. In this work, some existing surface flow models and subsurface flow models are reviewed; the interaction mechanisms of surface flow with subsurface porous media flow are discussed; numerical algorithms for solving coupled surface/subsurface flow models are proposed; in particular, preconditioning techniques and two grid algorithms are mathematically and numerically investigated.

In Chapter 1, we present some existing models for describing surface fluid flow motion as well as those for subsurface porous media flow motion. Moreover, we study some coupled models including both surface fluid flow and subsurface porous media flow. The interactions of fluid flow and porous media flow are reflected by transmission conditions at the interface between surface and subsurface. Other related multimodels for incompressible flows will also be introduced.

In Chapter 2, we focus on preconditioning techniques for the coupled Stokes/Darcy model, a linear model for the surface/subsurface flows coupling. Several decoupled preconditioners are proposed and analyzed. Especially, the convergence rate of GMRES method with these preconditioners is shown to be independent of meshsize. For improving the robustness with respect to physical parameters, coupled preconditioners are also theoretically and numerically investigated.

In Chapter 3, a two grid algorithm for decoupling the coupled Stokes/Darcy model is studied. The two grid algorithm consists of solving a coupled coarse grid problem, then solving two sub-problems in parallel. We use coarse grid solution to supplement boundary conditions at the interface for fine grid subproblems. Theoretical analysis shows that the two grid algorithm retains optimal approximation accuracy by choosing a proper scaling between coarse grid and fine grid. Both first order discretization and second order discretization are conducted to verify the theory. Moreover, we propose a multilevel algorithm based on the two grid technique. Numerical experiments show that our algorithms are very efficient and effective.

In Chapter 4, we propose several two grid algorithms for solving a coupled Navier-Stokes/Darcy model. Our two grid algorithms not only decouple the coupled problem but also linearize the nonlinear terms from Navier-Stokes equations and the interface conditions. Error analysis are given to show that our algorithms possess good approximation properties. Numerical justifications are also presented to show that our algorithms can greatly reduce the computational cost and can accurately approximate the solution of the coupled problem.

In Chapter 5, we study a linearized Navier-Stokes/Darcy coupling model. This model is extracted from coarse grid approximation and Picard iteration for solving the coupled nonlinear Navier-Stokes/Darcy problem. We adopt Green function based preconditioner for Oseen equations, combined with the preconditioning techniques for the coupled Stokes/Darcy model, to speed up the convergence rate of GMRES method. Comparisons with other preconditioners are numerically studied.

In the appendix, we review GMRES algorithm. Particularly, based on geometric properties of Krylov subspace methods, the convergence rate analysis of GMRES method is presented. Other issues including preconditioned GMRES algorithm and energy norm based GMRES algorithm will also be addressed.