Mortar Element Method For The Coupling Of Navier-Stokes And Darcy Flows

In this thesis,we investigate a mortar element method for the coupling of incom-pressible flow and porous media flow.The model is relevant to a variety of physical processes.The coupled system can be used to model the blood motion in the arte-rial vessel and to simulate surface water percolation through the riverbed filled by rocks and sands.Today,one serious environmental problem is groundwater pollution resulted by the transport of the contamination in rivers and lakes.The transport propagation can also be predicted by these coupling models.There are transport-diffusion phenomena in these cases.The coupled system consists of different partial differential equations where the free fluid is governed by Navier-Stokes equations and Darcy law can simulate the porous media.The fluid flow and porous media flow are coupled by such interface conditions:mass conservation(or called as flux continuity),the balance of the normal forces and the Beavers-Joseph-Saffman law which is the most accepted condition.We consider the dual-mixed formulation in Darcy region where velocity and pressure are both unknowns.The thesis consists of the following parts.Firstly,we study the nonlinear coupling system of Navier-Stokes and Daxcy flows.We present the mathematical expressions of the coupling model and two equivalent variational formulations.The difference is that whether there is a consistent term on the interface.Under the assumption on the viscosity and the right-hand terms,we prove the well-posedness of the weak formulation by virtue of Galerkin method and Brouwer fixed point theorem.We choose Taylor-Hood element to simulate Navier-Stokes equations and the lowest order Raviart-Thomas element to compute Darcy flow.In view of the fact that incompressible flow has a privileged role to play on the interface,the interface part of Navier-Stokes,boundary is chosen as the mortar.The opposite is non-mortar side.The mortar function space is defined to enforce the weak version of mass conservation on the interface which cannot guarantee the continuity of velocity.We prove the error estimation between the weak solutions and the numerical ones that is convergent of second order with respect to the mesh size of fluid region and is convergent of first order with respect to the mesh size in porous media.We adopt an iterative algorithm to solve the nonlinear scheme generated by the coupled system and prove that it will converge to the numerical solutions.In numerical experiments,there is a quadratic relationship between the mesh sizes of two regions.And for manufactured exact solutions,the numerical results is consistent with theoretical analysis.Secondly,we investigate a mortar element method for the time dependent cou-pling of Stokes and Darcy equations.All interface conditions hold for any time.We obtain the existence and the uniqueness of the weak solution by Galerkin method and the theory of ordinary differential equations.We choose Bernardi-Raugel ele-ment to simulate Stokes equation and the lowest order Raviart-Thomas element to solve Darcy flow.Therefore,the local flux continuity is guaranteed in each subregion.We adopt backward Euler method to obtain the fully discrete algorithm.Different from most summed results at time steps,we present the error estimation at each single step which is convergent of first order.We point out that the analysis is also suitable for other combinations of finite elements.And some numerical results axe provided to show the good performance of the developed algorithm in approximating the solution.It is obvious that the error between the manufactured exact solution and the numerical one linearly depends on the mesh size.Thirdly,we analyze the non-stationary coupling system-Navier-Stokes/Darcy.In the nonlinear system,there are partial derivatives of velocity with respect to time variable in both region.Since that most questions and difficulties have been solved in the above parts,this one is brief,relatively.Combining the results of the first and the second part,we establish the well-posedness of the weak formulation.The same finite element is chosen to solve the nonlinear coupling model as the second part.And the backward Euler method is also adopt for partial derivative with respect to time.Through a rigorous analysis,we prove the error.The proof is carried out by the operator which is induced by the mortar element solution of the stationary coupling of Navier-Stokes and Darcy flows in the first part.Upon the above results,we verify that it is effective to use the mortar elemen-t method combining different finite elements to solve the nonlinear and unsteady coupling problems.The proposed mortar element method allows non-matching grid which gives the flexibility in choosing finite element discretization according to cer-tain requirements in each subdomain.The proposed mortar element method not only inherits the advantages of different finite element methods in solving individual problems(Navier-Stokes or Darcy),but also has a weak(integral)expression of flux continuity on the interface.Due to finite elements used in both regions,the mor-tar element solution satisfies the local flux continuity.This property is critical for the coupling model with a transport equation because of avoiding the artificial mass sources.