The first chapter gave an overview of basic principles of fluid mechanics. In particular, it introduced the fundamental equations governing the motion of fluid flows. In this chapter we deal with approximation methods allowing these equations to be solved on a computer. We use a simple one-dimensional elliptic model problem in order to introduce the basic concepts. This problem will be carried throughout the chapter. Parabolic, hyperbolic, and multidimensional problems are treated in Chapters 3 and 4, respectively.

In Section 2.1 we begin with the derivation of a boundary-value problem (BVP) from a variational principle. Different types of boundary conditions are presented, with particular attention to the choice of test functions. The existence conditions for a solution are discussed within the Lax–Milgram theorem. Section 2.2 enlarges the scope to more general linear BVPs and introduces the approximation framework of Galerkin and collocation methods. In Section 2.3 we present various finite-element approximations, some of which have connections with spectral methods. Sections 2.4 and 2.5 discuss the spectral-element and the orthogonal collocation techniques, the two main forms of high-order methods studied in this book. Section 2.6 deals with the error estimation in connection with the various approximations introduced in the chapter. Section 2.7 is dedicated to some efficient solution techniques for the algebraic systems entailed by high-order methods. Finally, Section 2.8 discusses a numerical example.

In order to make the text reasonably self-contained, key mathematical notions have been gathered in two appendices.

Subdivision schemes are efficient computational methods for the design and representation of 3D surfaces of arbitrary topology. They are also a tool for the generation of refinable functions, which are instrumental in the construction of wavelets. This paper presents various flavours of subdivision, seasoned by the personal viewpoint of the authors, which is mainly concerned with geometric modelling. Our starting point is the general setting of scalar multivariate nonstationary schemes on regular grids. We also briefly review other classes of schemes, such as schemes on general nets, matrix schemes, non-uniform schemes and nonlinear schemes. Different representations of subdivision schemes, and several tools for the analysis of convergence, smoothness and approximation order are discussed, followed by explanatory examples.

Introduction

The first work on a subdivision scheme was by de Rahm (1956). He showed that the scheme he presented produces limit functions with a first derivative everywhere and a second derivative nowhere. The pioneering work of Chaikin (1974) introduced subdivision as a practical algorithm for curve design. His algorithm served as a starting point for extensions into subdivision algorithms generating any spline functions. The importance of subdivision to applications in computer-aided geometric design became clear with the generalizations of the tensor product spline rules to control nets of arbitrary topology. This important step has been introduced in two papers, by Doo and Sabin (1978) and by Catmull and Clark (1978). The surfaces generated by their subdivision schemes are no longer restricted to representing bivariate functions, and they can easily represent surfaces of arbitrary topology.

This article discusses finite element Galerkin schemes for a number of linear model problems in electromagnetism. The finite element schemes are introduced as discrete differential forms, matching the coordinate-independent statement of Maxwell's equations in the calculus of differential forms. The asymptotic convergence of discrete solutions is investigated theoretically. As discrete differential forms represent a genuine generalization of conventional Lagrangian finite elements, the analysis is based upon a judicious adaptation of established techniques in the theory of finite elements. Risks and difficulties haunting finite element schemes that do not fit the framework of discrete differential forms are highlighted.

Introduction

Most modern technology is inconceivable without harnessing electromagnetic phenomena. Hence the design and analysis of schemes for the approximate solution of electromagnetic field problems can claim a rightful place as a core discipline of numerical mathematics and scientific computing. However, for a long time it received far less attention among numerical analysts than, for instance, computational fluid dynamics and solid mechanics.

One reason might be that electromagnetism is described by a generically linear theory, in the sense that linear equations arise from basic physical principles. This is in stark contrast to continuum mechanics, where linear models only emerge through linearization of inherently nonlinear governing principles. Being linear, the fundamental laws of electromagnetism might have struck many mathematicians as ‘dull’. This view might also have been fostered by the misconception that electromagnetism basically boils down to plain second-order elliptic equations, which have been amply studied and are well understood.

An inverse eigenvalue problem concerns the reconstruction of a structured matrix from prescribed spectral data. Such an inverse problem arises in many applications where parameters of a certain physical system are to be determined from the knowledge or expectation of its dynamical behaviour. Spectral information is entailed because the dynamical behaviour is often governed by the underlying natural frequencies and normal modes. Structural stipulation is designated because the physical system is often subject to some feasibility constraints. The spectral data involved may consist of complete or only partial information on eigenvalues or eigenvectors. The structure embodied by the matrices can take many forms. The objective of an inverse eigenvalue problem is to construct a matrix that maintains both the specific structure as well as the given spectral property. In this expository paper the emphasis is to provide an overview of the vast scope of this intriguing problem, treating some of its many applications, its mathematical properties, and a variety of numerical techniques.

Introduction

In his book Finite-Dimensional Vector Spaces, Halmos (1974) wrote:

This interesting comment on the nomenclature of eigenvalue echoes the enigmatic yet important role that eigenvalues play in nature. One instance, according to Parlett (1998), is that ‘Vibrations are everywhere, and so too are the eigenvalues associated with them.’

This paper is concerned with the mathematical structure of the immersed boundary (IB) method, which is intended for the computer simulation of fluid–structure interaction, especially in biological fluid dynamics. The IB formulation of such problems, derived here from the principle of least action, involves both Eulerian and Lagrangian variables, linked by the Dirac delta function. Spatial discretization of the IB equations is based on a fixed Cartesian mesh for the Eulerian variables, and a moving curvilinear mesh for the Lagrangian variables. The two types of variables are linked by interaction equations that involve a smoothed approximation to the Dirac delta function. Eulerian/Lagrangian identities govern the transfer of data from one mesh to the other. Temporal discretization is by a second-order Runge–Kutta method. Current and future research directions are pointed out, and applications of the IB method are briefly discussed.

Introduction

The immersed boundary (IB) method was introduced to study flow patterns around heart valves and has evolved into a generally useful method for problems of fluid–structure interaction. The IB method is both a mathematical formulation and a numerical scheme. The mathematical formulation employs a mixture of Eulerian and Lagrangian variables. These are related by interaction equations in which the Dirac delta function plays a prominent role. In the numerical scheme motivated by the IB formulation, the Eulerian variables are defined on a fixed Cartesian mesh, and the Lagrangian variables are defined on a curvilinear mesh that moves freely through the fixed Cartesian mesh without being constrained to adapt to it in any way at all.

We give an overview of recent developments concerning the use of adjoint methods in two areas: the a posteriori error analysis of finite element methods for the numerical solution of partial differential equations where the quantity of interest is a functional of the solution, and superconvergent extraction of integral functionals by postprocessing.

Introduction

Output functionals

In many scientific and engineering applications that lead to the numerical approximation of solutions to partial differential equations, the objective is merely a rough, qualitative assessment of the details of the analytical solution over the computational domain, the quantitative concern being directed towards a few output functionals, derived quantities of particular engineering or scientific relevance.

For example, in aeronautical engineering, a CFD calculation of the flow around a transport aircraft at cruise conditions might be performed to investigate whether there are any unexpected shocks on the pylon connecting the engine to the wing, or whether there is an unexpected boundary layer separation caused by the main shock on the wing's suction surface. However, the engineer's overall concern is the impact of such phenomena on the lift and drag on the aircraft, and the quality of the CFD calculation is judged, first and foremost, by the accuracy of the lift and drag predictions. The fine details of the flow field are much less important, and are used only in a qualitative manner to suggest ways in which the design may be modified to improve the lift or drag.

We survey splitting methods for the numerical integration of ordinary differential equations (ODEs). Splitting methods arise when a vector field can be split into a sum of two or more parts that are each simpler to integrate than the original (in a sense to be made precise). One of the main applications of splitting methods is in geometric integration, that is, the integration of vector fields that possess a certain geometric property (e.g., being Hamiltonian, or divergence-free, or possessing a symmetry or first integral) that one wants to preserve. We first survey the classification of geometric properties of dynamical systems, before considering the theory and applications of splitting in each case. Once a splitting is constructed, the pieces are composed to form the integrator; we discuss the theory of such ‘composition methods’ and summarize the best currently known methods.