Stabilized Numerical Methods For The Three Kinds Of Problems Of Incompressible Fluid Flows

In this thesis,stabilized mixed finite element methods for the three kinds of in-compressible fluid flow problems are discussed.In the first kind,the Newtonian fluid flow is illustrated with the mixed finite element methods,where we have demonstrated the interaction between two incompressible fluid flow models.We consider two domains are coupled by a single interface,where each domain filled with the incompressible fluid governing by the Stokes model equations.To solve the coupled models,we have imposed the Nitsche's type interface conditions on the interface.We think that these interface conditions are more practical than the rest of the interface conditions.Indeed there is one drawback with the applicability of these conditions i.e.,some integrals on the inter-face are out of control in theoretical and numerical formulations.This is a challenging task in this Stokes-Stokes coupling model with the Nitsche's type interface conditions.We have solved this difficulty successfully.For an instant,to bound these inconsistent integrals on the interface of the two domains,some stabilization techniques are necessary to apply,which must ensure the continuity and uniqueness of the model equations.For this purpose,we have added the stabilization term and consistency term with the stabi-lization parameter in the discrete formulation.This stabilization term will never effect the original model,because of the mass conservation interface condition which make the stabilization term identically zero.We prove the stability of the proposed scheme in a consistent way by adopting the weak coercivity.For further accuracy,we also prove the optimal error estimates.To end this part,we perform several numerical simulations to check the efficiency and accuracy of the stabilized mixed finite element scheme for the Stokes-Stokes interface coupled fluid flow model.By which the optimal approximate accuracy are obtained by considering two different analytical solutions.Moreover,we use MINI elements and Taylor-Hood elements.The influence of the stabilization term and stabilization parameter are illustrated by a benchmark example known as lid-driven cavity flow.The applicability of the Stokes-Stokes multi-physics model is illustrated by introducing a conceptual domain.In the second kind of problem,a steady-state Oseen viscoelastic fluid flow obey-ing an Oldroyd-B type constitutive law is investigated.To study the viscoelastic fluid flow problems by mixed finite element methods,there are several methods applied in the literature.Among them,we concentrate two main methods in this thesis;the Dis-continuous Galerkin(DG)method and the streamline upwind Petrov-Galerkin(SUPG)method.We propose a stabilized scheme to solve the Oseen viscoelastic fluid flow with the lowest equal order finite elements and investigate both of the methods and their features in detail.The mixed formulation of Oseen viscoelastic fluid flow problems leads to three types of numerical instabilities.The first one is associated with the model equations instability which may face due to the stress,the second one is with the dominant of the convection,and the third one is about the suitable choice of the finite elements or polynomial spaces in the discrete formulation.The first two instability conditions are well described and analyzed in the existing finite element literature but so far the third instability condition is not considered rigorously.In this part,we have solved the viscoelastic fluid flow problem with the lowest equal order finite element methods.However,there are three unknowns in the viscoelastic fluid flow models;i.e.,velocity,pressure,and stress tensor respectively.To approximate unknown fields say the velocity,pressure and stress tensor are restricted to few finite elements in discrete formulation i.e.,Taylor-Hood and MINI elements.A stable numerical formulation is stable for only these finite element polynomial spaces which satisfy the well-known inf-sup or(LBB)condition.However,from the numerical point of view,these finite element spaces are limited and restrictive,sometimes they are not sufficient to approximate the optimal solutions,particularly when the problem needs to be solved in three dimensions.The alternative is to use other finite element spaces,we chose the lowest equal order finite element triples P1-P1-Pdc1 for discontinuous Galerkin(DG)triples and P1-P1-P1 triples streamline upwind petrove-Galerkin(SUPG)to approximate Oseen viscoelastic fluid flow model numerically.These finite elements are well-introduced and applied to find approximate solutions of the Navier-Stokes and Stokes model equations.It is therefore well known that these elements do not fulfil the stability or consistency(inf-sup)condition.Thus,owing to the violation of the essential stability condition,the system results instability in a weak solution or is not unique.To overcome this difficulty,a standard stabilization term is added in the discrete variational formulation.The technique applied herein possesses attractive features:it is penalized parameter-free,flexible in computation and does not require any higher-order derivatives.For the illustration of its accuracy,the stability and optimal error estimates are obtained.Three numerical benchmark tests are carried out to assess the stability and accuracy of the lowest-equal order mixed FE method.The third kind of problem aims to study the quasi-least square mixed finite element(FE)method for the linear and steady-state magnetohydrodynamic(MHD)equation-s.This method does not require(inf-sup)or LBB condition on the finite-dimensional subspaces.The quasi-least square finite element method has many advantages over the least square finite element method for nonlinear problems.The first one is that only L2-norm is used in these stabilized schemes so that they can be easily programmed.The second one is that by linearizing procedure one can derive simple iterative procedures with symmetric positive definite coefficient matrices.These simple iterative procedures are convergent in a large region of initial functions.The third one is quasi-least square finite element method and their convergence analysis are independent of the algorithm and the viscosity parameters.We investigate the existence of the solutions and obtain a priori error estimates.Finally,we give a numerical test for its convergence accuracy with true solutions.