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2021 | Buch

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Numerical Methods for Elliptic and Parabolic Partial Differential Equations

With contributions by Andreas Rupp

verfasst von: Prof. Dr. Peter Knabner, Prof. Dr. Lutz Angermann

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 graduate-level text provides an application oriented introduction to the numerical methods for elliptic and parabolic partial differential equations. It covers finite difference, finite element, and finite volume methods, interweaving theory and applications throughout. The book examines modern topics such as adaptive methods, multilevel methods, and methods for convection-dominated problems and includes detailed illustrations and extensive exercises. For students with mathematics major it is an excellent introduction to the theory and methods, guiding them in the selection of methods and helping them to understand and pursue finite element programming. For engineering and physics students it provides a general framework for the formulation and analysis of methods. This second edition sees additional chapters on mixed discretization and on generalizing and unifying known approaches; broader applications on systems of diffusion, convection and reaction; enhanced chapters on node-centered finite volume methods and methods of convection-dominated problems, specifically treating the now-popular cell-centered finite volume method; and the consideration of realistic formulations beyond the Poisson's equation for all models and methods.

Inhaltsverzeichnis

Frontmatter

Chapter 0. For Example: Modelling Processes in Porous Media with Differential Equations

Abstract

This chapter illustrates the scientific context in which differential equation models may occur, in general, and also in a specific example. Section 0.1 reviews the fundamental equations, for some of them discretization techniques will be developed and investigated in this book. In Sections 0.2–0.4 we focus on reaction and transport processes in porous media. These sections are independent of the remaining parts and may be skipped by the reader. Section 0.5, howevershould be consulted because it fixes some notation to be used later on.

Peter Knabner, Lutz Angermann

Chapter 1. For the Beginning: The Finite Difference Method for the Poisson Equation

Abstract

In this section we want to introduce the finite difference method, frequently abbreviated as FDM, using the Poisson equation on a rectangle as an example. By means of this example and generalizations of the problem, advantages and limitations of the approach will be elucidated. Also, in the following section the Poisson equation will be the main topic, but then on an arbitrary domain.

Peter Knabner, Lutz Angermann

Chapter 2. The Finite Element Method for the Poisson Equation

Abstract

The finite element method, frequently abbreviated by FEM, was developed in the fifties in the aircraft industry, after the concept had been independently outlined by mathematicians at an earlier time.

Peter Knabner, Lutz Angermann

Chapter 3. The Finite Element Method for Linear Elliptic Boundary Value Problems of Second Order

Abstract

We now continue the definition and analysis of the “correct” function spaces that we began in (2.19)–(2.22).

Peter Knabner, Lutz Angermann

Chapter 4. Grid Generation and A Posteriori Error Estimation

Abstract

An essential step in the implementation of the finite element method (and also of the finite volume method as described in Chapter 8) is to create an initial “geometric discretization” of the domain $\Omega .$ This part of a finite element code is usually included in the so-called preprocessor (see also Section 2.4.1). In general, a stand-alone finite element code consists further of the intrinsic kernel (assembling of the finite-dimensional system of algebraic equations, rearrangement of data (if necessary), solution of the algebraic problem) and the postprocessor (editing of the results, extraction of intermediate results, preparation for graphic output, a posteriori error estimation).

Peter Knabner, Lutz Angermann

Chapter 5. Iterative Methods for Systems of Linear Equations

Abstract

We consider again the system of linear equations with nonsingular matrix $A \in \mathbb {R}^{m,m}$, right-hand side $b \in \mathbb {R}^m$, and solution $x\in \mathbb {R}^m$. As shown in Chapters 2 and 3, such systems of equations arise from finite element discretizations of elliptic boundary value problems. The matrix A is the stiffness matrix and thus sparse, as can be seen from (2.41).

Peter Knabner, Lutz Angermann

Chapter 6. Beyond Coercivity, Consistency, and Conformity

Until now, the general framework has been as follows: Let V be a Hilbert space with the scalar product $\langle \cdot , \cdot \rangle _{V}$ and induced norm $\Vert \cdot \Vert _{V}$, $a:\; V \times V \rightarrow \mathbb {R}$ a bilinear form, and $\ell \in V'$.

Peter Knabner, Lutz Angermann

Chapter 7. Mixed and Nonconforming Discretization Methods

The FEM studied in Chapter 3 and the node-oriented FVM in Chapter 8 are all conforming with a variational formulation based on $V\subset H^1(\Omega )$ and thus exhibit inter-element continuity (see Theorem 3.21).

Peter Knabner, Lutz Angermann

Chapter 8. The Finite Volume Method

Over the last decades, finite volume methods (FVMs) have enjoyed great popularity in various fields of computational mathematics (for example, in computational aerodynamics, fluid mechanics, solid-state physics, semiconductor device modelling...). In comparison with the standard discretization tools, namely finite element methods and finite difference methods, FVMs are occupying, in some sense, an intermediate position. Actually, treatments in the context of generalized finite difference methods as well as the finite element approach can be found in the literature.

Peter Knabner, Lutz Angermann

Chapter 9. Discretization Methods for Parabolic Initial Boundary Value Problems

Abstract

In this section, initial boundary value problems for the linear case of the differential equation (0.33) are considered. We choose the form (3.13) together with the boundary conditions (3.19)–(3.21), which have already been discussed in Section 0.5.

Peter Knabner, Lutz Angermann

Chapter 10. Discretization Methods for Convection-Dominated Problems

Abstract

As we have seen in the introductory Chapter 0, the modelling of transport and reaction processes in porous media results in differential equations of the form

$$ \partial _t u - \nabla \cdot (\boldsymbol{K}\nabla u-\boldsymbol{c}u) = f\,, $$

which is a special case of the form (0.33), and just the time-dependent version of the formulation in divergence form (3.36) with $b=1$.

Peter Knabner, Lutz Angermann

Chapter 11. An Outlook to Nonlinear Partial Differential Equations

Abstract

Despite the concentration of most of the book on linear boundary value problems and their discretizations, many real-world models are rather nonlinear elliptic or parabolic boundary value problems with various types of nonlinearities. To be specific, we restrict ourselves to the formulation in divergence form (3.36) and its time-dependent counterpart and the classical FEM based on Chapter 2 and Chapter 3. Let $a, \ell $ be the forms from (3.36) to (3.39), a first nonlinear (here: semilinear) boundary value problem reads.

Peter Knabner, Lutz Angermann

Backmatter

Titel: Numerical Methods for Elliptic and Parabolic Partial Differential Equations
verfasst von: Prof. Dr. Peter Knabner
Prof. Dr. Lutz Angermann
Verlag: Springer International Publishing
Electronic ISBN: 978-3-030-79385-2
Print ISBN: 978-3-030-79384-5
DOI: https://doi.org/10.1007/978-3-030-79385-2