Stabilized Mixed Finite Element Methods For The Multi-Physics Fluid Flow Models

In this thesis,we investigate several multi-physics interface systems for the surface and subsurface flow through stabilized mixed finite element methods in mathematical and numerical points of view.The conduit region is null or void and filled with fluid,which moves freely can be modeled by the Stokes or Navier-Stokes equations.On the other hand,the porous medium is defined as a material volume consisting of the solid matrix with the interconnected void,and it is mainly characterized by its porosity,i.e.,the ratio of the void or null space to the total volume of the region.Hence the subsurface flow for the porous medium can be governed by Darcy's law or dual-porosity and dual-permeability equations.The dual-porosity and dual-permeability equations describe the heterogeneous properties of two coexisting,transmittable and contiguous porous medium,known as microfracture and matrix region.To couple two separate models across the interface,we require physically valid effective and efficient coupling conditions.The continuous fluid communication is considered between the microfractures and conduit region,which is modeled by three traditional interface conditions named as the conservation of mass,the balance of normal forces and Beavers-Joseph-Saffman coupling conditions.On the other hand,the no-communication interface condition describes the no-direct fluid-transfer between the matrix block and the conduit region.To solve these multi-physics models,we develop new mixed stabilized finite element techniques.The mixed finite element discretization for the subsurface flow in the porous medium describes the macroscopic properties of the filtration process,which is convenient to use because of its superior conservation properties and convenience of the computation of flux.The robust mixed finite element methods developed here do not require any Lagrange multiplier,but a mesh dependent stabilization term with a penalty/relaxation parameter is introduced essentially to ensure the numerical stability of the algorithms.We develop a stabilized mixed finite element algorithm for the stationary Stokesdual-permeability fluid flow model.A detail derivation of continuity and weak coercivity are reported.The optimal error estimate is derived.An iterative algorithm is developed,and the convergence of the iterative scheme to the coupled scheme is derived.On the other hand,two stabilized mixed finite element methods are developed to study the evolutionary dual-porosity-Stokes fluid flow model.The first approach is to propose and analyze a coupled stabilized mixed finite element method to lay a solid ground.Then in order to significantly improve the computational efficiency,a decoupled stabilized mixed finite element method is constructed by using two-level decoupling idea.The stability and error analysis are derived for two methods.To study the non-stationary Stokes-Darcy system,we propose a new technique.In tradition,the error equations are constructed from the difference between the variational formulation and finite element discretized scheme.Herein,we construct the exact solution directly from the model problem by considering test functions in finite element spaces and then introduce the error equations.Coupled and decoupled schemes are proposed by assuming a stabilization term and consistency term.Moreover,we propose a coupled mathematical model for the closed-loop geothermal system,which mainly consists of a network of underground heat exchange pipelines to extract the heat from the geothermal reservoir.The new mathematical model considers the heat transfer between two different flow regions,namely the porous media flow in the geothermal reservoir and the free flow in the pipes.Darcy's law and Navier-Stokes equations are considered to govern the flows in these two regions with Boussinesq approximation,respectively,while the heat equation is coupled with the flow equations to describe the heat transfer in both regions.Furthermore,on the interface between the two regions,four physically valid interface conditions are considered to describe the continuity of the temperature and the heat flux as well as the no-fluid-communication feature of the closed-loop geothermal system.To solve the proposed model accurately and efficiently,we develop a stabilized decoupled finite element method which decouples not only the two flow regions but also the heat field and the flow field in each region.The stability of the proposed method is proved.The exclusive features of the numerical methods and applicability of the multi-physics models are illustrated through several pseudo-realistic numerical experiments.