Continuous Interior Penalty Finite Element Methods For Flow Problems

With the rapid development of digital computers, numerical simulations for flow problems have become an indispensable tool in biology, medicine, chemistry, engineering, environmental sciences and many other fields. During the last three decades, various numerical methods, such as finite difference, finite element, finite volume and spectral methods, have been proposed and studied in the literature. Among these methods, the finite element method is the most powerful and versatile one due to its flexibility in handling the complex geometry and its high accuracy. However, the development of finite element methods for flow problems that are accurate, efficient, and robust remains a challenging problem.As is well known, the standard Galerkin approximation for flow prob-lems may suffer from numerical instabilities arising from two main sources. The first is due to the presence of advection terms in the governing equa-tions, which can result in spurious node-to-node oscillations primarily in the velocity field. Such oscillations become more apparent for high Reynolds number flows and flows with sharp layers in the solution. The second is due to the stiff velocity-pressure coupling, which excludes the use of sim-ple low-order velocity-pressure pairs. However, from an engineering point of view, it is preferable to use the equal-order pairs of mixed finite elements for the velocity and pressure since it is computationally convenient in a parallel processing and multigrid context. In order to overcome both difficulties, a possible remedy is to introduce some additional stabilization terms into the corresponding variational formu-lation. The classical stabilization techniques are of Petrov-Galerkin (SUPG) type or weighted least squares (GLS) type. These methods stabilize a prob-lem through the introduction of residual-based terms. It has been shown that these methods stabilize both instabilities due to dominating convection and instabilities due to the velocity/pressure coupling. Despite the success of these classical stabilization approaches in theory and applications, one can find in recent papers a critical evaluation of these approaches, see[14,23,60]. A fundamental drawback of the residual-based methods is that various terms need to be added to the weak formulation in order to guarantee the consis-tency of the methods in a strong way. This is especially troublesome for complex flows with compressible features or with some additional variables as in reactive flow problems. Furthermore, an extension to time-dependent problems is problematic since a space-time, finite element approach has to be employed. For a discussion of such problems, we refer to [14].Recently, the continuous interior penalty (CIP) finite element method of Douglas and Dupont [45] have emerged as an alternative. The idea is to add a term penalizing the jump of the solution gradient over element faces as a uni-fied treatment of all the above mentioned instabilities. The key observation is that the stabilization operators may be used to control all the nonsymmetric first order terms of flow problems and that they give control only of the part of the operator that is not in the finite element spaces. In this sense, the CIP method is a minimal stabilized method as proposed in [18]. The analysis for high Peclet number problems was given by Burman and Hansbo in [25], and inf-sup stability for Stokes systems using equal order interpolation was proven in [26]. In [23], Burman, Fernandez and Hansbo provided an exten-sion to Oseen's equations by obtaining estimates independent of the local Reynolds number. Subsequently, Bonito and Burman studied the Oldroyd-B model of viscoelastic flows in [11]. Moreover, it was shown in [19] how this method provides a natural link between conforming and nonconforming stabilized finite element methods.Compared to the classical stabilization method, the CIP methods have several advantages, mainly thanks to the fact that the stabilization term does not couple to all the terms of the residual, and thus is independent of both time derivatives, source terms and higher derivatives. As a consequence, space and time discretization commute and the method can be combined with any type of time discretization. Moreover, the stabilization term of the CIP methods is symmetric, thereby making it attractive for optimization problems and solver aspects [13]. Another important feature of this approach is that the stabilization parameter is independent of the diffusion coefficient. This will be particularly important when dealing with nonlinear problems where this coefficient depends on the discrete solution.The purpose of the present dissertation is to introduce and analyze the continuous interior penalty finite element formulations for several flow problems:Sobolev equations, Stokes equations, Darcy-Stokes equations and Navier-Stokes equations. Continuous piecewise polynomial finite elements are used to approximate each component of the unknowns. We follow the framework proposed in [19,23] using weakly imposed boundary conditions as introduced by Nitsche (see [55]). Our analysis and numerical results show that the CIP methods are very effective for such problems.The dissertation is divided into four chapters.In chapter1, we consider the Sobolev equations with convection-dominated term. Equations of this type arise in the flow of fluids through fissured rock [7], thermodynamics [36] and other applications. For a discussion of existence and uniqueness results, see [41,47]. In many applications, it is necessary to consider Sobolev equations with convection-dominated term. The goal of this chapter is to present implicit and semi-implicit time-stepping methods for continuous interior penalty (CIP) finite element approximations of such equations. Stability is obtained by adding an interior penalty term giving L2-control of the jump of the gradient over clement faces. Several (?)-stable time-stepping methods arc analyzed and shown to be unconditionally stable and optimally convergent. We show that the contribution from the gradient jumps leading to an extended matrix pattern may be extrapolated from pre-vious time steps, and hence handled explicitly without loss of stability and accuracy.In chapter2, we investigate the velocity-prcssure-gradient, pscudostress-velocity, and velocity-stress-rotation formulations of Stokes flow. Nowadays, the numerical simulation of incompressible Newtonian flows is an important subtask in many industrial applications and remains within the focus of in-tensive scholar research. Over the years, much effort has been devoted to the construction of finite element methods for the Stokes equations based on the velocity-pressure formulation or its variants. However, in many applica-tions, such as turbulent or non-Newtonian flows, the velocity gradient or the stress tensor is actually the more relevant variable. Hence, it is natural to consider a formulation containing the velocity gradient or the stress tensor as a fundamental unknown. This avoids degrading of accuracy, which is in-evitable in the process of numerical differentiation when the velocity gradient is computed by taking derivatives of the velocity. Here, we consider the CIP methods for numerically solving these three formulations of the Stokes equa-tions. The methods use continuous pieccwise polynomials of degree k≥1for all the components of the approximate solutions. It is shown that these equal-order finite clement pairs are stable and yield quasi-optimal accuracy for sufficiently smooth solutions.In chapter3, we study the Darcy-Stokes problem. The development of effient numerical methods for the coupled Darcy and Stokes flows problem has become a very active research area due to its significance in hydrology, environment science and biofluid dynamics. These problems are modeled by the Darcy equations in the porous medium and the Stokes equations in the fluid region. The interface conditions consist of mass conservation, balance of normal forces and the Beavers-Joseph-Saffman law. The construction of finite element methods for this coupled model that are robust and accurate is not trivial, because the Darcy and Stokes solutions have very different regularity properties and the tangential velocity may be discontinuous on the interface between the two regions. In this chapter, we consider a porous medium entirely enclosed within a fluid region and present a continuous in-terior penalty finite element method for the corresponding coupled problem. The method uses standard continuous piecewise-linear elements for veloc-ities and pressures in both domains, and piecewise-constant or continuous piccewise-linear elements for the Lagrange multiplier. Meshes do not need to match at the interface. We show stability, convergence and a priori error estimates for the associated Galerkin scheme.In chapter4, we consider a fully discrete stabilized finite element method for the two-dimensional time-dependent Navicr-Stokes equations, where the time discretization is based on the Euler implicit/semi-explicit scheme and the Euler implicit/explicit scheme, and the spatial discretization is based on the CIP finite element method. The method uses standard continuous polynomial finite element spaces for the velocity and the pressure. Inf-sup stability and stability for convection-dominated flows are obtained by adding a term giving L2-control of the jump of the solution gradient over clement edges. We prove a priori error estimates independent of the local Reynolds number and give some numerical results to illustrate the performance of our methods.