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2021 | Buch

Lebesgue spaces Kapitel lesen Erstes Kapitel lesen

Finite Elements I

Approximation and Interpolation

verfasst von: Prof. 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 first 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 I is divided into 23 chapters plus two appendices on Banach and Hilbert spaces and on differential calculus. This volume focuses on the fundamental ideas regarding the construction of finite elements and their approximation properties. It addresses the all-purpose Lagrange finite elements, but also vector-valued finite elements that are crucial to approximate the divergence and the curl operators. In addition, it also presents and analyzes quasi-interpolation operators and local commuting projections. The volume starts with four chapters on functional analysis, which are packed with examples and counterexamples to familiarize the reader with the basic facts on Lebesgue integration and weak derivatives. Volume I also reviews important implementation aspects when either developing or using a finite element toolbox, including the orientation of meshes and the enumeration of the degrees of freedom.

Inhaltsverzeichnis

Frontmatter

Elements of functional analysis

Frontmatter
1. Lebesgue spaces
Abstract
The objective of the four chapters composing Part I of this book is to recall (or gently introduce) some elements of functional analysis that will be used throughout the book: Lebesgue integration, weak derivatives, and Sobolev spaces. We focus in this chapter on Lebesgue integration and Lebesgue spaces. Most of the results are stated without proof, but they are illustrated through various examples.
Alexandre Ern, Jean-Luc Guermond
2. Weak derivatives and Sobolev spaces
Abstract
We investigate the notion of differentiation for Lebesgue integrable functions. We introduce an extension of the classical concept of derivative and partial derivative which is called weak derivative. This notion is used throughout the book. We also introduce the concept of Sobolev spaces. These spaces are useful to study the well-posedness of partial differential equations and their approximation using finite elements.
Alexandre Ern, Jean-Luc Guermond
3. Traces and Poincaré inequalities
Abstract
This chapter reviews two types of results on the Sobolev spaces introduced in the previous chapter. The first one concerns the notion of trace (loosely speaking, the boundary values) of functions in Sobolev spaces. The second one is about functional inequalities due to Poincaré and Steklov. The validity of some of these results relies on smoothness properties of the boundary of the set on which the functions are defined. We mainly focus on Lipschitz sets in this book.
Alexandre Ern, Jean-Luc Guermond
4. Distributions and duality in Sobolev spaces
Abstract
The dual space of a Sobolev space is not only composed of functions (defined almost everywhere), but it also contains more sophisticated objects called distributions which are defined by their action on smooth functions with compact support. Dual Sobolev spaces are useful to handle singularities on the right-hand side of PDEs. They are also useful to give a meaning to the tangential and the normal traces of vector-valued fields that are not smooth enough to admit traces as members of a suitable Sobolev space. The extension of the notions of tangential and normal traces is done in this case by invoking integration by parts formulas involving the curl or the divergence operators.
Alexandre Ern, Jean-Luc Guermond

Introduction to finite elements

Frontmatter
5. Main ideas and definitions
Abstract
This chapter introduces key notions for finite elements, such as degrees of freedom, shape functions, and interpolation operator. These notions are illustrated on Lagrange finite elements and modal finite elements, for which the degrees of freedom are values at specific nodes and moments against specific test functions, respectively.
Alexandre Ern, Jean-Luc Guermond
6. One-dimensional finite elements and tensorization
Abstract
This chapter presents important examples of finite elements, first in dimension one, then in multiple dimensions using tensor-product techniques. Important computational issues related to the manipulation of high-order polynomial bases are addressed. We also show how to approximate integrals over intervals using the roots of the Legendre and Jacobi polynomials.
Alexandre Ern, Jean-Luc Guermond
7. Simplicial finite elements
Abstract
This chapter deals with finite elements defined on a simplex (triangle in 2D, tetrahedron in 3D). The degrees of freedom are either nodal values at some points on the simplex or integrals over the faces or the edges of the simplex, and the associated functional space is composed of multivariate polynomials of prescribed total degree. We focus our attention on scalar-valued finite elements. The results extend to the vector-valued case by reasoning componentwise.
Alexandre Ern, Jean-Luc Guermond

Finite element interpolation

Frontmatter
8. Meshes
Abstract
We study how to build a mesh of a bounded domain, i.e., a finite collection of cells forming a partition of the domain. Building a mesh is the first important task to realize when one wants to approximate some PDEs posed in the domain. The viewpoint we adopt in this book is that each mesh cell is the image of a reference cell by some smooth diffeomorphism that we call geometric mapping. We show how to construct the geometric mapping and we present various important notions concerning meshes. We also discuss mesh-related data structures and mesh generators.
Alexandre Ern, Jean-Luc Guermond
9. Finite element generation
Abstract
We show how to generate a finite element in each mesh cell from a reference finite element. To this purpose, we need one new concept in addition to the geometric mapping: a functional transformation that maps functions defined on the current mesh cell to functions defined on the reference cell. Key examples of such transformations are the Piola transformations. These transformations arise naturally in the chain rule when one investigates how the standard differential operators (gradient, curl, divergence) are transformed by the geometric mapping. The construction presented in this chapter provides the cornerstone for the analysis of the finite element interpolation error.
Alexandre Ern, Jean-Luc Guermond
10. Mesh orientation
Abstract
Orienting the edges and the faces of a mesh is crucial when working with finite elements whose degrees of freedom invoke normal or tangential components of vector fields. This notion is also important when working with high-order scalar-valued finite elements, since it helps to enumerate consistently all the degrees of freedom in the mesh cells sharing an edge or a face with other mesh cells. In this chapter, we focus on matching meshes, and we assume that the meshes are affine. We first explain how to orient meshes. Then we introduce the important notion of generation-compatible orientation. Finally, we investigate whether simplicial, quadrangular, and hexahedral meshes can be equipped with a generation-compatible orientation.
Alexandre Ern, Jean-Luc Guermond
11. Local interpolation on affine meshes
Abstract
We analyze the local finite element interpolation error for smooth functions. We restrict the material to affine meshes and to relatively simple functional transformations. We introduce the notion of shape-regular families of affine meshes, we study the transformation of Sobolev norms, and we present important approximation results collectively known as the Bramble–Hilbert lemmas. The main result proved in this chapter is an upper bound on the local interpolation error over each mesh cell for smooth functions.
Alexandre Ern, Jean-Luc Guermond
12. Local inverse and functional inequalities
Abstract
Inverse inequalities rely on the fact that all the norms are equivalent in finite-dimensional normed vector spaces. The term ‘inverse’ refers to the fact that high-order Sobolev (semi)norms are bounded by lower-order (semi)norms, but the constants involved in these estimates either tend to zero or to infinity as the meshsize goes to zero. Our purpose is then to study how the norm-equivalence constants depend on the local meshsize and the polynomial degree of the reference finite element. We also derive some local functional inequalities valid in infinite-dimensional spaces. All these inequalities are regularly invoked in this book. In the whole chapter, we consider a shape-regular sequence of affine meshes.
Alexandre Ern, Jean-Luc Guermond
13. Local interpolation on nonaffine meshes
Abstract
In this chapter, we extend the results of Chapter 11 to nonaffine meshes. For simplicity, the functional transformation is the pullback by the geometric mapping, but this mapping is now nonaffine. The first difficulty consists of comparing Sobolev norms. This is not a trivial task since the chain rule involves higher-order derivatives of the geometric mapping. The second difficulty is to define a notion of shape-regularity for mesh sequences built using nonaffine geometric mappings. We show how to do this using a perturbation theory, and we present various examples.
Alexandre Ern, Jean-Luc Guermond
14. finite elements
Abstract
The goal of this chapter is to construct vector-valued finite elements to approximate fields with integrable divergence. The finite elements introduced in this chapter can be used, e.g., to approximate Darcy’s equations which constitute a fundamental model for porous media flows. The focus here is on defining a reference element and generating finite elements on the mesh cells. The estimation of the interpolation error is done in Chapters 16 and 17. We detail the construction for the simplicial Raviart–Thomas finite elements. Some alternative finite elements are outlined at the end of the chapter.
Alexandre Ern, Jean-Luc Guermond
15. finite elements
Abstract
The goal of this chapter is to construct vector-valued finite elements to approximate fields with integrable curl. The finite elements introduced in this chapter can be used, e.g., to approximate (simplified forms of) Maxwell’s equations which constitute a fundamental model in electromagnetism. The focus here is on defining a reference element and generating finite elements on the mesh cells. The interpolation error analysis is done in Chapters 16 and 17. We detail the construction for the simplicial Nédélec finite elements of the first kind. Some alternative elements are outlined at the end of the chapter.
Alexandre Ern, Jean-Luc Guermond
16. Local interpolation in and (I)
Abstract
In this chapter and the next one, we study the interpolation operators associated with the finite elements introduced in Chapters 14 and 15. We consider a shape-regular sequence of affine simplicial meshes with a generation-compatible orientation. In the present chapter, we show how the degrees of freedom attached to the faces and the edges can be extended to Sobolev spaces with enough smoothness. On the way, we discover fundamental commuting properties of the interpolation operators embodied in the de Rham complex.
Alexandre Ern, Jean-Luc Guermond
17. Local interpolation in and (II)
Abstract
In this chapter, we continue our investigation of the interpolation operators associated with the finite elements introduced in Chapters 14 and 15. We consider a shape-regular sequence of affine simplicial meshes with a generation-compatible orientation. The key idea here is to extend the degrees of freedom on the faces and the edges by requiring some integrability of the divergence or the curl of the function to be interpolated. This approach is useful when such integrability properties can be extracted from a PDE solved by the function in question, as it is often the case in applications (e.g., for Darcy’s equations and for Maxwell’s equations). The crucial advantage of the present approach over that from the previous chapter based on the scale of Sobolev spaces is that interpolation error estimates with lower smoothness requirements can be obtained. On the way, we also devise a face-to-cell lifting operator that is useful in the analysis of nonconforming approximations of elliptic problems.
Alexandre Ern, Jean-Luc Guermond

Finite element spaces

Frontmatter
18. From broken to conforming spaces
Abstract
We introduce broken Sobolev spaces and broken finite element spaces based on a mesh. Then we identify jump conditions across the mesh interfaces that are necessary and sufficient for every function in some broken Sobolev space to have an integrable gradient, curl, or divergence. These conditions lead to the notion of conforming finite element spaces. Finally, we show how to construct \(L^1\)-stable (local) interpolation operators in the broken finite element space with optimal local approximation properties.
Alexandre Ern, Jean-Luc Guermond
19. Main properties of the conforming subspaces
Abstract
We continue the study of the interpolation properties of the conforming finite element subspaces introduced in Chapter 18. These spaces are defined from a broken finite element space by requiring that some jumps across the mesh interfaces are zero. The cornerstone of the present construction, which is presented in a unified way, is a connectivity array with ad hoc clustering properties of the local degrees of freedom. In the present chapter, we postulate the existence of the connectivity array and show how it allows us to build global shape functions and a global interpolation operator in the conforming finite element space. The actual construction of this mapping is done in Chapters 20 and 21. We assume that the mesh is matching for simplicity.
Alexandre Ern, Jean-Luc Guermond
20. Face gluing
Abstract
The goal of this chapter and the following one is to construct the connectivity array introduced in the previous chapter. In this chapter, we see how to enforce the zero-jump condition across every mesh interface by means of the degrees of freedom on the two mesh cells sharing this interface. In particular, we identify two key structural assumptions on the finite element making this construction possible. The first assumption is called face unisolvence, and the second one is called face matching. We first introduce these ideas with Lagrange elements to make the argumentation easier to understand. Then we generalize the concepts to the Nédélec and the Raviart–Thomas finite elements in a unified setting that encompasses all the finite elements considered in the book.
Alexandre Ern, Jean-Luc Guermond
21. Construction of the connectivity classes
Abstract
In this chapter, we finish the construction of the connectivity classes which we characterize by means of an equivalence relation. We show that the resulting equivalence classes verify the two key assumptions introduced in Chapter 19. Our starting point is to assume that the finite element at hand satisfies the two fundamental assumptions introduced in Chapter 20: the face unisolvence assumption and the face matching assumption. These two assumptions turn out to be sufficient to fully characterize the connectivity classes of Raviart–Thomas elements. For the other elements (Lagrange, canonical hybrid, and Nédélec) for which there are degrees of freedom attached to geometric entities of smaller dimension, we have to consider two additional abstract assumptions, the M-unisolvence assumption and the M-matching assumption, which we show hold true for these elements. At the end of the chapter, we propose enumeration techniques that facilitate the practical construction of the connectivity array.
Alexandre Ern, Jean-Luc Guermond
22. Quasi-interpolation and best approximation
Abstract
One of the objectives of this chapter is to estimate the decay rate of the best-approximation errors of functions in Sobolev spaces by members of conforming finite element spaces. The interpolation operators constructed so far do not give a satisfactory answer to the above question when the functions have a low smoothness index. In this chapter, we introduce the important notion of quasi-interpolation, i.e., we build linear operators that are \(L^1\)-stable, are projections onto conforming finite element spaces, and have optimal local approximation properties. We do this by composing one of the \(L^1\)-stable operators onto the larger broken finite element space with a simple averaging operator. We also adapt the construction to enforce zero traces at the boundary. We finally study the approximation properties of the \(L^2\)-orthogonal projection onto the conforming finite element spaces. The material of this chapter is important to investigate the approximation of solutions to PDEs with low regularity.
Alexandre Ern, Jean-Luc Guermond
23. Commuting quasi-interpolation
Abstract
The quasi-interpolation operators introduced in Chapter 22 are \(L^1\)-stable, are projections and have optimal (local) approximation properties. However, they do not commute with the usual differential operators (gradient, curl, and divergence), which makes them difficult to use to approximate simultaneously a vector-valued function and its curl or its divergence. Since these commuting properties are important in some applications, we introduce in this chapter quasi-interpolation operators that are \(L^1\)-stable, are projections, have optimal (global) approximation properties, and have the expected commuting properties. The key idea is to compose the canonical interpolation operators with mollification operators, i.e., smoothing operators based on the convolution with a smooth kernel.
Alexandre Ern, Jean-Luc Guermond
Backmatter
Metadaten
Titel
Finite Elements I
verfasst von
Prof. Alexandre Ern
Jean-Luc Guermond
Copyright-Jahr
2021
Electronic ISBN
978-3-030-56341-7
Print ISBN
978-3-030-56340-0
DOI
https://doi.org/10.1007/978-3-030-56341-7