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In this text, we introduce the basic concepts for the numerical modelling of partial differential equations. We consider the classical elliptic, parabolic and hyperbolic linear equations, but also the diffusion, transport, and Navier-Stokes equations, as well as equations representing conservation laws, saddle-point problems and optimal control problems. Furthermore, we provide numerous physical examples which underline such equations. In particular, we discuss the algorithmic and computer implementation aspects and provide a number of easy-to-use programs.

The text does not require any previous advanced mathematical knowledge of partial differential equations: the absolutely essential concepts are reported in a preliminary chapter. It is therefore suitable for students of bachelor and master courses in scientific disciplines, and recommendable to those researchers in the academic and extra-academic domain who want to approach this interesting branch of applied mathematics.



1. A brief survey on partial differential equations

The purpose of this chapter is to recall the basic concepts related to partial differential equations (PDE, in short). For a wider coverage see [RR04], [Eva98], [LM68], [Sal08].

2. Elements of functional analysis

In this chapter, we recall a number of concepts used extensively in this textbook: functional and bilinear forms, distributions, Sobolev spaces, L p spaces. For a more in-depth reading, the reader can refer to e.g. [Sa108],[Yos74], [Bre86], [LM68], [Ada75].

3. Elliptic equations

This chapter is devoted to the introduction of elliptic problems and to their weak formulation. Although our introduction is quite basic, the complete novice to functional analysis is invited to consult Chap. 2 before reading it.

4. The Galerkin finite element method for elliptic problems

In this chapter, we describe the numerical solution of the elliptic boundary-value problems considered in Chap. 3 by introducing the Galerkin method. We then illustrate the finite element method as a particular case. The latter will be further developed in the following chapters.

5. Parabolic equations

In this chapter, we consider parabolic equations of the form
$$ \frac{{\partial u}} {{\partial t}} + Lu = f, x \in \Omega , t > 0, $$
where Ω is a domain of ℝd, d = 1, 2, 3, f = f(x, t) is a given function, L = L(x) is a generic elliptic operator acting on the unknown u = u(x,t). When solved only for a bounded temporal interval, say for 0 < t < T, the region QT = Ω × (0, T) is called cylinder in the space ℝd × ℝ+ (see Fig. 5.1). In the case where T = +∞, Q={(x,t):x ∈ Ω, t > 0} will be an infinite cylinder.

6. Generation of 1D and 2D grids

As we have seen, the finite element method for the solution of partial differential equations requires a “triangulation” of the computational domain, i.e. a partition of the domain in simpler geometric entities (for instance, triangles or quadrangles in two dimensions, tetrahedra, prisms or hexahedra in three dimensions), called the elements, which verify a number of conditions. Similar partitions stand at the base of other approximation methods, such as the finite volume method (see Chap. 9) and the spectral element method (see Chap. 10). The set of all elements is the so-called computational grid (or, simply, grid, or mesh).

7. Algorithms for the solution of linear systems

In this chapter we make a quick and elementary introduction of some of the basic algorithms that are used to solve a system of linear algebraic equations. For a more thorough presentation we advise the reader to refer to, e.g., [QSS07], Chap. 3 and 4, [Saa96]and[vdV03].

8. Elements of finite element programming

In this chapter, we focus more deeply on a number of aspects relating to the translation of the finite element method into computer code. This implementation process can hide some pitfalls. Beyond the syntactic requirements of a given programming language, the need for a high computational efficiency implementation requires a codification that is generally not the immediate translation of what has been seen during theoretical presentation. Efficiency depends on many factors, including the language used and the architecture on which one works. Personal experience can play a role as fundamental as learning from a textbook. Moreover, although spending time searching for a bug in the code or for a more efficient data structure can sometimes appear to be a waste of time, it (almost) never is. For this reason, we wish to propose the present chapter as a sort of “guideline” for trials that the reader can perform on his own, rather that a chapter to be studied in the traditional sense.

9. The finite volume method

The finite volume method is a very popular method for the space discretization of partial differential problems in conservation form. For an in-depth presentation of the method, we suggest the monographs [LeV02a] and [Wes01].

10. Spectral methods

As we have seen in Chap. 4, when we approximate boundary-value problems using the finite element method, the order of convergence is anyhow limited by the degree of the polynomials used, also in the case where solutions are very regular. In this chapter, we will introduce spectral methods, for which the convergence rate is only limited by the regularity of the solution of the problem (and is exponential for analytical solutions). For a detailed analysis we refer to [CHQZ06, CHQZ07, Fun92, BM92].

11. Diffusion-transport-reaction equations

In this chapter, we consider problems of the following form
$$ \left\{ \begin{gathered} - div(\mu \nabla u) + b \cdot \nabla u + \sigma u = f in\Omega , \hfill \\ u = 0 on\partial \Omega , \hfill \\ \end{gathered} \right. $$
where µ, σ, f and b are given functions (or constants). In the most general case, we will suppose that µ ∈ L(Ω) with µ(x) ≥ µ0 > 0, σ ∈ L2(Ω) with σ(x) ≥ 0 a.e. in Ω,b ∈ [L(Ω)]2 withdiv(b) ∈ L2(Ω), and f ∈ L2(Ω).

12. Finite differences for hyperbolic equations

In this chapter, we will deal with time-dependent problems of hyperbolic type. For their derivation and for an in-depth analysis see e.g. [Sal08], Chap. 4. We will limit ourselves to considering the numerical approximation using the finite difference method, which was historically the first one to be applied to this type of equations. To introduce in a simple way the basic concepts of the theory, most of our presentation will concern problems depending on a single space variable. Finite element approximations will be addressed in Chap. 13, the extension to nonlinear problems in Chap. 14.

13. Finite elements and spectral methods for hyperbolic equations

In this chapter, we will illustrate how to apply Galerkin methods, and in particular the finite element method and the spectral one, to the spatial and/or temporal discretization of scalar hyperbolic equations. We will treat both the continuous as well as discontinuous finite element cases.

14. Nonlinear hyperbolic problems

In this chapter, we introduce some examples of nonlinear hyperbolic problems. We will point out some characteristic properties of such problems, the most relevant being their possibility to generate discontinuous solutions also in the case of continuous initial and boundary data. The numerical approximation of these problems is an all but simple task. Here, we will simply limit ourselves to point out how finite difference and finite element schemes can be applied in the case of one-dimensional equations.

15. Navier-Stokes equations

Navier-Stokes equations describe the motion of a fluid with constant density ρ in a domain Ω ⊂ ℝd (with d = 2, 3). They write as follows
$$ \left\{ \begin{gathered} \tfrac{{\partial u}} {{\partial t}} - div[v(\nabla u + \nabla u^T )] + (u \cdot \nabla )u + \nabla p = f, x \in \Omega ,t > 0, \hfill \\ divu = 0, x \in \Omega ,t > 0, \hfill \\ \end{gathered} \right. $$
u being the fluid velocity, p the pressure divided by the density (which will simply be called “pressure”), \( \nu = \tfrac{\mu } {\rho } \) the kinematic viscosity, µ the dynamic viscosity, and f a forcing term per unit mass that we suppose to belong to the space L2(ℝ+; [L2(Ω)]d) (see Sec. 5.2). The first equation is that of conservation of linear momentum, the second one that of conservation of mass, which is also called the continuity equation. The term (u ⋅ ∇)u describes the process of convective transport, while —div [υ(∇u + ∇uT)] the process of molecular diffusion. System (15.1) can be derived by the analogous system for compressible flows introduced in Chap. 14 by assuming ρ constant, using the continuity equation (that in the current assumption takes the simplified form divu = 0) to simplify the various terms, and finally dividing the equation by ρ. Note that in the incompressible case (15.2) the energy equation has disappeared. Indeed, even though such an equation can still be written for incompressible flows, its solution can be carried out independently once the velocity field is obtained from the solution of (15.1).

16. Optimal control of partial differential equations

In this chapter we will introduce the basic concepts of optimal control for linear elliptic partial differential equations. At first we present the classical theory in functional spaces “à la J.L.Lions”, see [Lio71] and [Lio72]; then we will address the methodology based on the use of the Lagrangian functional (see, e.g., [Mau81], [BKR00] and [Jam88]). Finally, we will show two different numerical approaches for control problems, based on the Galerkin finite element method.

17. Domain decomposition methods

In this chapter we will introduce the domain decomposition method (DD, in short). In its most common version, DD can be used in the framework of any discretization method for partial differential equations (such as, e.g. finite elements, finite volumes, finite differences, or spectral element methods) to make their algebraic solution more efficient on parallel computer platforms. In addition, DD methods allow the reformulation of any given boundary-value problem on a partition of the computational domain into subdomains. As such, it provides a very convenient framework for the solution of heterogeneous or multiphysics problems, i.e. those that are governed by differential equations of different kinds in different subregions of the computational domain.

18. Reduced basis approximation for parametrized partial differential equations

In this chapter we describe the basic ideas of reduced basis (RB) approximation methods for rapid and reliable evaluation of input-output relationships in which the output is expressed as a functional of a field variable that is the solution of an input-parametrized partial differential equation (PDE). We shall focus on linear output functionals and affinely parametrized linear elliptic coercive PDEs; however the methodology is much more generally applicable as we discuss in Sec. 18.6 at the end of this chapter. The combination with an efficient a posteriori error estimation is a key factor for RB methods to be computationally successful.


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