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Finite Elements III

First-Order and Time-Dependent PDEs

verfasst von: Prof. Dr. Alexandre Ern, Jean-Luc Guermond

Verlag: Springer International Publishing

Buchreihe : Texts in Applied Mathematics

Enthalten in: Springer Professional "Wirtschaft+Technik" , Springer Professional "Technik" , Springer Professional "Wirtschaft"

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Über dieses Buch

This book is the third volume of a three-part textbook suitable for graduate coursework, professional engineering and academic research. It is also appropriate for graduate flipped classes. Each volume is divided into short chapters. Each chapter can be covered in one teaching unit and includes exercises as well as solutions available from a dedicated website. The salient ideas can be addressed during lecture, with the rest of the content assigned as reading material. To engage the reader, the text combines examples, basic ideas, rigorous proofs, and pointers to the literature to enhance scientific literacy.

Volume III is divided into 28 chapters. The first eight chapters focus on the symmetric positive systems of first-order PDEs called Friedrichs' systems. This part of the book presents a comprehensive and unified treatment of various stabilization techniques from the existing literature. It discusses applications to advection and advection-diffusion equations and various PDEs written in mixed form such as Darcy and Stokes flows and Maxwell's equations. The remainder of Volume III addresses time-dependent problems: parabolic equations (such as the heat equation), evolution equations without coercivity (Stokes flows, Friedrichs' systems), and nonlinear hyperbolic equations (scalar conservation equations, hyperbolic systems). It offers a fresh perspective on the analysis of well-known time-stepping methods. The last five chapters discuss the approximation of hyperbolic equations with finite elements. Here again a new perspective is proposed. These chapters should convince the reader that finite elements offer a good alternative to finite volumes to solve nonlinear conservation equations.

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Inhaltsverzeichnis

Frontmatter

First-order PDEs

Frontmatter

Chapter 56. Friedrichs’ systems

Abstract

In this chapter, we introduce the prototypical model problem for first-order PDEs: it is a system of first-order linear PDEs introduced in 1958 by Friedrichs. This system enjoys symmetry and positivity properties and, in the literature, it is often referred to as Friedrichs’ system. Friedrichs’ formalism is very powerful and encompasses several model problems. Important examples are the advection-reaction equation, the div-grad problem related to Darcy’s equations, and the curl-curl problem related to Maxwell’s equations. This theory will be used systematically in the following chapters.

Alexandre Ern, Jean-Luc Guermond

Chapter 57. Residual-based stabilization

Abstract

This chapter is concerned with the approximation of Friedrichs’ systems using \(H^1\)-conforming finite elements. The main issue one faces in this context is to achieve stability. The stabilization techniques presented in this chapter are inspired by the least-squares (LS), or minimal residual, technique from linear algebra. The LS approximation gives optimal error estimates in the graph norm, but unfortunately it gives suboptimal \(L^2\)-error estimates in most situations. The Galerkin/least-squares (GaLS) method improves the situation by combining the standard Galerkin approach with the LS technique and mesh-dependent weights. The GaLS method gives quasi-optimal \(L^2\)-error estimates and optimal mesh-dependent graph-norm estimates. We also show that the GaLS method can be combined with a boundary penalty technique to enforce boundary conditions weakly.

Alexandre Ern, Jean-Luc Guermond

Chapter 58. Fluctuation-based stabilization (I)

Abstract

In this chapter, we still use \(H^1\)-conforming finite elements and a boundary penalty technique, but we consider a different stabilization technique. One motivation is that the residual-based stabilization from the previous chapter is delicate to use when approximating time-dependent PDEs since the time derivative is part of the residual. The techniques devised in this chapter and the next one avoid this difficulty. The starting observation is that \(H^1\)-conforming test functions cannot control the gradient of \(H^1\)-conforming functions since the gradient generally exhibits jumps across the mesh interfaces. The idea behind fluctuation-based stabilization is to gain full control on the gradient by adding a least-squares penalty on the part of the gradient departing from the \(H^1\)-conforming space, and this part can be viewed as a fluctuation. Stabilization techniques based on this idea include the continuous interior penalty (CIP) method, studied in this chapter, and two-scale stabilization techniques such as the local projection stabilization (LPS) and the subgrid viscosity (SGV) methods, which are studied in the next chapter. We present in this chapter a unified analysis based on an abstract set of assumptions. We show in this chapter and the next one how to satisfy these assumptions using CIP, LPS, and SGV.

Alexandre Ern, Jean-Luc Guermond

Chapter 59. Fluctuation-based stabilization (II)

Abstract

In this chapter, we continue the unified analysis of fluctuation-based stabilization techniques for Friedrichs’ systems. We now focus on two closely related stabilization techniques known in the literature as local projection stabilization (LPS) and subgrid viscosity (SGV). The key idea is to introduce a two-scale decomposition of the discrete \(H^1\)-conforming finite element space which leads to the notions of resolved and fluctuating (or subgrid) scales. Both stabilization techniques rely on a least-squares penalty: LPS penalizes the fluctuation of the gradient and SGV penalizes the gradient of the fluctuation. As for the CIP technique studied in the previous chapter, we verify that the abstract design conditions introduced in the previous chapter are met with LPS and SGV.

Alexandre Ern, Jean-Luc Guermond

Chapter 60. Discontinuous Galerkin

Abstract

In this chapter, instead of using stabilized \(H^1\)-conforming finite elements, we consider the discontinuous Galerkin (dG) method. The stability and convergence properties of the method rely on choosing a numerical flux across the mesh interfaces. Choosing the centered flux yields suboptimal convergence rates for smooth solutions. The stability properties of the method are tightened by penalizing the interface jumps, which corresponds to upwinding in the case of advection-reaction equations. This method gives the same error estimates as those obtained with stabilized \(H^1\)-conforming finite elements. Here again, the boundary conditions are enforced by a boundary penalty technique.

Alexandre Ern, Jean-Luc Guermond

Chapter 61. Advection-diffusion

Abstract

In this chapter, we want to solve a model problem where the PDE comprises a first-order differential operator modeling advection processes and a second-order term modeling diffusion processes. The difficulty in approximating an advection-diffusion equation can be quantified by the Péclet number which is equal to the meshsize times the advection velocity divided by the diffusion coefficient. When the Péclet number is large, the standard Galerkin approximation is plagued by spurious oscillations. These oscillations disappear if very fine meshes are used, but a more effective approach using coarser meshes is to resort to stabilization. In this chapter, we focus on the Galerkin/least-squares (GaLS) stabilization, but any stabilized \(H^1\)-conforming method or the dG method can also be used.

Alexandre Ern, Jean-Luc Guermond

Chapter 62. Stokes equations: Residual-based stabilization

Abstract

Employing inf-sup stable mixed finite elements to solve Stokes-like problems may seem to be a cumbersome constraint. The goal of this chapter is to show that it is possible to work with pairs of finite elements that do not satisfy the inf-sup condition provided the Galerkin formulation is slightly modified. This is done by extending to the Stokes problem the stabilization techniques that have been presented in the previous chapters. Although all these techniques can be adapted to the Stokes problem, for brevity we only exemplify three of them. We focus on the Galerkin/least-squares method (GaLS) in this chapter. The continuous interior penalty and the discontinuous Galerkin methods are investigated in the next chapter.

Alexandre Ern, Jean-Luc Guermond

Chapter 63. Stokes equations: Other stabilizations

Abstract

We continue in this chapter the study of stabilization techniques to approximate the Stokes problem with finite element pairs that do not satisfy the inf-sup condition. We now focus our attention on the continuous interior penalty and the discontinuous Galerkin methods.

Alexandre Ern, Jean-Luc Guermond

Parabolic PDEs

Frontmatter

Chapter 64. Bochner integration

Abstract

The goal of this chapter is to introduce a mathematical setting to formulate parabolic problems in some weak form. The viewpoint we are going to take is to consider functions defined on a bounded time interval with values in some Banach (or Hilbert) space composed of functions defined on the space domain. The key notions we develop in this chapter are the Bochner integral and the weak time derivative of functions that are Bochner integrable.

Alexandre Ern, Jean-Luc Guermond

Chapter 65. Weak formulation and well-posedness

Abstract

The goal of this chapter is to derive a weak formulation of a model parabolic problem and to establish its well-posedness. The prototypical example is the heat equation. To this purpose, we use the Bochner integration theory presented in the previous chapter.

Alexandre Ern, Jean-Luc Guermond

Chapter 66. Semi-discretization in space

Abstract

We are concerned in this chapter with the semi-discretization in space of the model parabolic problem introduced in the previous chapter. We use conforming finite elements for the space approximation. Error estimates are derived by invoking coercivity-like arguments. Semi-discretization in space leads to a (large) system of coupled ordinary differential equations (ODEs). This system of ODEs can then be discretized in time by many time-stepping techniques, as exemplified in the following chapters. This approach is often called method of lines in the literature.

Alexandre Ern, Jean-Luc Guermond

Chapter 67. Implicit and explicit Euler schemes

Abstract

The goal is now to discretize in time the space semi-discrete parabolic problem considered in the previous chapter. Since this problem is a system of coupled (linear) ODEs, its time discretization can be done by using one of the numerous time-stepping techniques available from the literature. In this chapter, we focus on the implicit (or backward) Euler scheme and on the explicit (or forward) Euler scheme, which are both first-order accurate in time. Second-order implicit schemes called BDF2 and Crank–Nicolson are investigated in the next chapter. The standard viewpoint in the literature is to interpret the above schemes as finite differences in time. This is the perspective we adopt in this chapter and the next one. We broaden the perspective later on by introducing a discrete space-time formulation and by considering higher-order time discretization methods.

Alexandre Ern, Jean-Luc Guermond

Chapter 68. BDF2 and Crank–Nicolson schemes

Abstract

In this chapter, we discuss two time-stepping techniques that deliver second-order accuracy in time and, like the implicit Euler method, are unconditionally stable. One technique is based on the second-order backward differentiation formula (BDF2), and the other, called Crank–Nicolson, is based on the midpoint quadrature rule. Since the BDF2 method is a two-step scheme, it is not well suited to time step adaptation. Moreover, the stability analysis must account for the way the scheme is initialized at the first time step (we use here an implicit Euler step). In contrast to the BDF2 time stepping, the Crank–Nicolson scheme, like the implicit Euler scheme, is a one-step method. We will see however that the stability properties of the Crank–Nicolson method are not as strong as those of the implicit Euler method.

Alexandre Ern, Jean-Luc Guermond

Chapter 69. Discontinuous Galerkin in time

Abstract

In the previous two chapters, we have used finite differences to approximate the time derivative in the space semi-discrete parabolic problem. We now adopt a different viewpoint directly relying on a space-time weak formulation. The time approximation is realized by using piecewise polynomial functions over the time mesh. The test functions are discontinuous at the time nodes, thereby allowing for a time-stepping process, i.e., the discrete formulation decouples into local problems over each time step. This leads to two new families of schemes. In the present chapter, we study the discontinuous Galerkin method in time, where the trial functions are also discontinuous at the time nodes. In the next chapter, we study the continuous Petrov–Galerkin methods where the trial functions are continuous. The lowest-order version of the discontinuous Galerkin technique is the implicit Euler scheme, and the lowest-order version of the Petrov–Galerkin technique is the Crank–Nicolson scheme. All these schemes are implicit Runge–Kutta methods.

Alexandre Ern, Jean-Luc Guermond

Chapter 70. Continuous Petrov–Galerkin in time

Abstract

In this chapter, we continue the study started in the previous chapter on higher-order time approximation schemes using a space-time functional framework. The trial functions are now continuous in time and piecewise polynomials with a polynomial degree that is one order higher than that of the test functions. The resulting technique is called continuous Petrov–Galerkin method, and its lowest-order version is the Crank–Nicolson scheme. Like the discontinuous Galerkin schemes, the continuous Petrov–Galerkin schemes are implicit one-step methods. They can also be interpreted as implicit Runge–Kutta methods.

Alexandre Ern, Jean-Luc Guermond

Chapter 71. Analysis using inf-sup stability

Abstract

In this chapter, we revisit the well-posedness of the model parabolic problem, i.e., we give another proof of Lions’ theorem using the framework of the BNB theorem. In other words, we establish the well-posedness by proving an inf-sup condition. Then we exploit the inf-sup condition to revisit the stability and the error analysis for various approximation techniques investigated in the previous chapters: (1) the space semi-discrete problem; (2) the implicit Euler scheme; (3) the discontinuous Galerkin scheme; (4) the continuous Petrov–Galerkin scheme.

Alexandre Ern, Jean-Luc Guermond

Time-dependent Stokes equations

Frontmatter

Chapter 72. Weak formulations and well-posedness

Abstract

This chapter focuses on the weak formulation of the time-dependent Stokes equations. We consider two possible weak formulations. The first one enforces the divergence-free constraint on the velocity field without introducing the pressure. This formulation can be handled by using the same analysis tools as for parabolic problems. The second weak formulation includes the pressure. This formulation entails some subtleties concerning the smoothness in time of the pressure and of the time derivative of the velocity. Both formulations hinge on the Bochner integration theory.

Alexandre Ern, Jean-Luc Guermond

Chapter 73. Monolithic time discretization

Abstract

The present chapter deals with the approximation of the time-dependent Stokes equations. We use stable mixed finite elements for the space discretization in a conforming setting. The time discretization can be done with any of the techniques considered for the heat equation. For brevity, we focus on the implicit Euler scheme and on higher-order implicit Runge–Kutta (IRK) schemes. The discretization process gives at each time step a saddle point problem coupling the velocity and the pressure, so that the linear algebra is in general more involved than when dealing with the heat equation.

Alexandre Ern, Jean-Luc Guermond

Chapter 74. Projection methods

Abstract

This chapter gives a brief overview of some splitting techniques to approximate the time-dependent Stokes problem in time. The common feature of the algorithms is that each time step leads to subproblems where the velocity and the pressure are uncoupled. The linear algebra resulting from the space approximation is therefore simplified, making these methods attractive for their efficiency. In this chapter, we review a class of techniques known in the literature as projection methods where the accuracy in time is limited to second order. The algorithms reviewed in the next chapter are based on an artificial compressibility perturbation of the mass conservation equation and can reach arbitrary accuracy in time.

Alexandre Ern, Jean-Luc Guermond

Chapter 75. Artificial compressibility

Abstract

In this chapter, we study a time-stepping technique for the time-dependent Stokes equations based on an artificial compressibility perturbation of the mass conservation equation. This technique presents some advantages with respect to the projection methods. It avoids solving a Poisson equation at each time step, and it can be extended to high order in a rather straightforward manner.

Alexandre Ern, Jean-Luc Guermond

Time-dependent first-order linear PDEs

Frontmatter

Chapter 76. Well-posedness and space semi-discretization

Abstract

In this chapter, we consider time-dependent Friedrichs’ systems. The prototypical example is the linear transport equation. We first derive a functional setting for the model problem and establish its well-posedness. For simplicity, we assume that the differential operator in space is time-independent. Then we construct a space semi-discretization of the problem using stabilized finite elements. We focus on fluctuation-based stabilization techniques.

Alexandre Ern, Jean-Luc Guermond

Chapter 77. Implicit time discretization

Abstract

In this chapter, we continue the study of the time-dependent Friedrichs’ systems. In the previous chapter, we have established the well-posedness of the continuous model problem and we have discretized the problem in space using \(H^1\)-conforming finite elements, a boundary penalty technique, and fluctuation-based stabilization. In this chapter, we now discretize the problem in time and focus on the implicit Euler scheme.

Alexandre Ern, Jean-Luc Guermond

Chapter 78. Explicit time discretization

Abstract

In this chapter, we consider the same space semi-discrete problem as in the previous chapter, but we now discretize it in time using an explicit scheme. We first discuss generic properties of explicit Runge–Kutta schemes (ERK). Then we analyze the explicit Euler scheme, second-order two-stage ERK schemes, and third-order three-stage ERK schemes. The key advantage of explicit schemes over implicit schemes is that the linear algebra at each time step is greatly simplified since one has to invert only the mass matrix. However, the stability of ERK schemes requires that the time step be limited by a CFL-like condition.

Alexandre Ern, Jean-Luc Guermond

Nonlinear hyperbolic PDEs

Frontmatter

Chapter 79. Scalar conservation equations

Abstract

This chapter gives a brief description of the theory of scalar conservation equations. We introduce the notions of weak and entropy solutions and state existence and uniqueness results. Even if the initial data is smooth, the solution of a generic scalar conservation equation may lose smoothness in finite time, and weak solutions are in general nonunique. Uniqueness is recovered by enforcing constraints that are called entropy conditions. We finish this chapter by exploring the structure of a one-dimensional Cauchy problem called Riemann problem where the initial data is composed of two constant states. Understanding the structure of the solution to the Riemann problem is important to understand the approximation techniques discussed later on.

Alexandre Ern, Jean-Luc Guermond

Chapter 80. Hyperbolic systems

Abstract

The objective of this chapter is to introduce the concept of hyperbolic systems and to generalize the notions introduced in the previous chapter to this class of equations. The novelty here is that the notion of maximum principle is no longer valid and is replaced by the concept of invariant sets.

Alexandre Ern, Jean-Luc Guermond

Chapter 81. First-order approximation

Abstract

This chapter focuses on the approximation of nonlinear hyperbolic systems using finite elements. We describe a somewhat loose adaptation to finite elements of a scheme introduced by Lax. The method, introduced by Guermond, Nazarov, and Popov, can be informally shown to be first-order accurate in time and space and to preserve every invariant set of the hyperbolic system. The time discretization is based on the forward Euler method and the space discretization employs finite elements. The theory applies regardless of whether \(H^1\)-conforming or discontinuous elements are used.

Alexandre Ern, Jean-Luc Guermond

Chapter 82. Higher-order approximation

Abstract

The objective of this chapter is to describe techniques that preserve the invariant domain property of the algorithm introduced in the previous chapter and increase its accuracy in time and space. The argumentation for the time approximation is done for general hyperbolic systems, but the argumentation for the space approximation is done for scalar conservation equations only. The general situation is treated in the next chapter.

Alexandre Ern, Jean-Luc Guermond

Chapter 83. Higher-order approximation and limiting

Abstract

The objective of this chapter is to describe techniques for the solution of hyperbolic systems that are at least (informally) second-order accurate in space and invariant domain preserving. As seen in the previous chapter, one can make the method more accurate in space by decreasing the first-order graph viscosity. Another technique, which is very efficient when working with nonsmooth data or with solutions with shocks or contact discontinuities, consists of using the consistent mass matrix instead of the lumped mass matrix in the approximation of the time derivative. These two techniques increase the accuracy in space but deliver an update that can step out of the invariant domain. We then show that this defect can be corrected by applying a conservative convex limiting technique.

Alexandre Ern, Jean-Luc Guermond

Backmatter

Titel: Finite Elements III
verfasst von: Prof. Dr. Alexandre Ern
Jean-Luc Guermond
Copyright-Jahr: 2021
Verlag: Springer International Publishing
Electronic ISBN: 978-3-030-57348-5
Print ISBN: 978-3-030-57347-8
DOI: https://doi.org/10.1007/978-3-030-57348-5

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