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Über dieses Buch

This book simultaneously presents the theory and the numerical treatment of elliptic boundary value problems, since an understanding of the theory is necessary for the numerical analysis of the discretisation. It first discusses the Laplace equation and its finite difference discretisation before addressing the general linear differential equation of second order. The variational formulation together with the necessary background from functional analysis provides the basis for the Galerkin and finite-element methods, which are explored in detail. A more advanced chapter leads the reader to the theory of regularity. Individual chapters are devoted to singularly perturbed as well as to elliptic eigenvalue problems. The book also presents the Stokes problem and its discretisation as an example of a saddle-point problem taking into account its relevance to applications in fluid dynamics.

Inhaltsverzeichnis

Frontmatter

Chapter 1. Partial Differential Equations and Their Classification Into Types

Abstract
Abstract This chapter introduces the partial differential equations and their distinction into three types.
Wolfgang Hackbusch

Chapter 2. The Potential Equation

Abstract
In Section 2.1 the simplest but prototypical elliptic differential equation of second order is presented. The solutions of this equation are called harmonic. Together with a boundary condition, one obtains a boundary-value problem. An important tool is the singularity function, which is defined in Section 2.2. The Green formulae allow a representation of the solution in Theorem 2.8. In Section 2.3 functions with mean-value property are introduced. It is shown that these functions coincide with harmonic functions. The mean-value property implies the maximum minimum principle: non-constant functions have no local extrema. An important conclusion is the uniqueness of the solution (Theorem 2.18). Finally, in Section 2.4, it is shown that the solution depends continuously on the boundary data.
Wolfgang Hackbusch

Chapter 3. The Poisson Equation

Abstract
In Section 3.1 the Poisson equation –Δu=f is introduced, and the uniqueness of the solution is proved. The Green function is defined in Section 3.2. It allows the representation (3.6) of the solution, provided it is existing. Concerning the existence, Theorem 3.13 contains a negative statement (cf. Section 3.3): The Poisson equation with a continuous right-hand side f may possess no classical solution. A sufficient condition for a classical solution is the Hölder continuity of f as stated in Theorem 3.18. Section 3.4 introduces Green’s function for the ball. In the two-dimensional case, Riemann’s mapping theorem allows the construction of the Green function for a large class of domains. In Section 3.5 we replace the Dirichlet boundary condition by the Neumann condition. The final Section 3.6 is a short introduction into the integral equation method. The solution of the boundary-value problem can indirectly be obtained by solving an integral equation.
Wolfgang Hackbusch

Chapter 4. Difference Methods for the Poisson Equation

Abstract
The difference method replaces the derivatives by difference quotients. Section 4.1 describes the difference method applied to the one-dimensional Poisson equation \(-u^{\prime\prime}\;=\;f\). This simple example is chosen to show the generation of the discrete system of equations. The difference equations are complemented by the Dirichlet boundary condition. The equations of the resulting linear system correspond to the inner grid points, while the boundary data appear in the right– hand side of the system.
Wolfgang Hackbusch

Chapter 5. General Boundary-Value Problems

Abstract
Section 5.1 introduces the general elliptic linear differential equation of second order together with the Dirichlet boundary values. An important statement is the maximum-minimum principle in §5.1.2. In §5.1.3 sufficient conditions for the uniqueness of the solution and the continuous dependence on the data are proved. The discretisation of the general differential equation in a square is described in §5.1.4. Section 5.2 treats alternative boundary conditions replacing the Dirichlet data. Examples are the Neumann condition, the conormal derivative and the Robin boundary condition. Their discretisation (cf. §5.2.2) for general domains is rather laborious. Section 5.3 discusses differential equations of higher order. In particular, the biharmonic equation of fourth order is described in §5.3.1 followed by equations of order 2m in §5.3.2. The discretisation of the biharmonic equation is in §5.3.3.
Wolfgang Hackbusch

Chapter 6. Tools from Functional Analysis

Abstract
Here we collect those definitions and statements which are needed in the next chapters. Section 6.1 introduces the normed spaces, Banach and Hilbert spaces as well as the operators as linear and bounded mappings between these spaces. In most of the later applications these spaces will be function spaces, containing for instance the solutions of the differential equations. It will turn out that the Sobolev spaces from Section 6.2 are well suited for the solutions of boundary value problems. The Sobolev space \(H^{k}(\it\Omega)\) and \(H^{k}_{0}(\it\Omega)\) for nonnegative integers k as well as \(H^{s}(\it\Omega)\) for real \(s\geq 0\) are introduced. The definition of the trace (restriction to the boundary Γ) will be essential in §6.2.5 for the interpretation of boundary values. To this end the Sobolev spaces \(H^{s}(\it\Gamma)\) of functions on the boundary Γ must be defined (cf. Theorem 6.57). Sobolev’s Embedding Theorem 6.48 connects Sobolev spaces and classical spaces. Section 6.3 introduces dual spaces and dual mappings. Compactness properties are important for statements about the unique solvability. Compact operators and the Riesz–Schauder theory are presented in Section 6.4. The weak formulation \(H^{s}(\it\Gamma)\) of the boundary-value problem is based on bilinear forms described in Section 6.5. The inf-sup condition in Lemma 6.94 is a necessary and sufficient criterion for the solvability of the weak formulation.
Wolfgang Hackbusch

Chapter 7. Variational Formulation

Abstract
Techniques based on classical function spaces are less suited for proving the existence of a solution of a boundary-value problem. Section 7.1 introduces another approach via a variational problem (Dirichlet’s principle). Combining the variational formulation with the Sobolev spaces will be successful. In Section 7.2 the boundary-value problem of order 2m with homogeneous Dirichlet conditions is transferred into the variational formulation in the space \(H^{m}_{0}(\it\Omega)\). Existence of a solution in \(H^{m}_{0}(\it\Omega)\) follows in Theorem 7.8 from the \(H^{m}_{0}(\it\Omega)\)-ellipticity which is discussed, e.g., in the Theorems 7.3 and 7.7. In Section 7.3 we consider inhomogeneous Dirichlet boundary-value problems. The natural boundary condition in Section 7.4 follows from variation in \(H^{m}_{0}(\it\Omega)\) without any restrictions. In the case of the Poisson equation one obtains the Neumann condition, in the general case the conormal boundary derivative appears. We investigate how general boundary conditions can be formulated as variational problem. Complications appearing for differential equations of higher order are explained by taking the example of the biharmonic equation.
Wolfgang Hackbusch

Chapter 8. The Finite-Element Method

Abstract
In Chapter 7 the variational formulation has been introduced to prove the existence of a (weak) solution. Now it will turn out that the variational formulation is extremely important for numerical purposes. It establishes a new, very flexible discretisation method. After historical remarks in Section 8.1 we introduce the Ritz–Galerkin method in Section 8.2. The basic principle is the replacement of the function space V in the variational formulation by an N-dimensional space. This leads to a system of N linear equations (§8.2.1). As described in §8.2.2, the theory from Chapter 7 can be applied. In §8.2.3 two criteria, the inf-sup condition and V-ellipticity are described which are sufficient for solvability. §8.2.4 contains numerical examples. Error estimates are discussed in Section 8.3. The quasioptimality of the Ritz–Galerkin method proved in §8.3.1 shifts the discussion to the approximation properties of the subspace (§8.3.2). The finite elements introduced in Section 8.4 form a special finite-dimensional subspace offering many practical advantages. The corresponding error estimates are given in Section 8.5. Generalisations to differential equations of higher order and to non-polygonal domains are investigated in Section 8.6. An important practical subject are a-posteriori error estimates discussed in Section 8.7. When solving the arising system of linear equations, the properties of the system matrix is of interest which are investigated in Section 8.8. Several other topics are sketched in the final Section 8.9.
Wolfgang Hackbusch

Chapter 9. Regularity

Abstract
The previous results can only guarantee the existence of weak solutions, i.e., in the case of m = 1 only first derivatives in \(L^2(\it\Omega)\) can be proven. In the beginning we also asked for second derivatives satisfying the equation \(\mathcal{O}(h^k)\) requires a solution in \(H^{1+k}(\it\Omega)\). Therefore the crucial question is, under what conditions the weak solution also belongs to Sobolev spaces of higher order (cf. Section 9.1). Section 9.2 characterises a specific property of elliptic solutions: In the interior of the domain the solution is smoother than close to the boundary. In the case of analytic coefficients the solution is also analytic in the interior and the bounds of the (higher) derivatives improve with the distance from the boundary. This behaviour also holds for the singularity and Green’s function. In Section 9.3 the regularity properties of solutions of difference schemes is studied. When comparing the error estimates for difference methods in §4.5 with those for finite-element estimates in §8.5 one observes that the latter require much weaker smoothness of the solution. However, one gets similar estimate for difference methods if one uses suitable discrete regularity properties (cf. §9.3.3). Unfortunately, the proof of these properties is rather technical, much more involved, and inflexible compared with the finite-element case.
Wolfgang Hackbusch

Chapter 10. Special Differential Equations

Abstract
If the boundary-value problems have special properties, one often uses special discretisations for them. We give two examples. In Section 10.1 the principal part has jumping coefficients. Starting from the variational formulation, one obtains a strong formulation for each subdomain in which the coefficients are smooth. In addition, one gets transition equations at the inner boundary Υ. Finite-element methods should use a triangulation which follows Υ. Finally, in §10.1.4, we discuss the case that coefficients of terms different from the principal part are discontinuous. Typically the differential operators in fluid dynamics are nonsymmetric because of a derivative of first order. If this convections term becomes dominant, we obtain a singularly perturbed problem which is discussed in Section 10.2. In this case other discretisation variants are appropriate. In the case of difference method there is a conflict between stability and consistency conditions. Usual finite-element discretisation have similar difficulties. A remedy is the streamline-diffusion method explained in §10.2.3.2.
Wolfgang Hackbusch

Chapter 11. Elliptic Eigenvalue Problems

Abstract
If \((L-\lambda I)u\;=\;f\) is not solvable, the Riesz–Schauder theory states that there are eigenvalues and eigenvectors. The weak formulation of the eigenvalue problem and some basic terms are discussed in Section 11.1. Since we do not require the system to be symmetric, also the adjoint problem must be treated. Section 11.2 is devoted to the finite-element discretisation by a family \(\{V_h\;:\;h \in\;H\}\) of subspaces. Theorems 11.13 and 11.15 state an important result: Each eigenvalue λ 0 of L is associated to a sequence of discrete eigenvalues converging to λ 0, and vice versa. The corresponding error estimates are given in §11.2.3 for the case of simple eigenvalues. A related estimate for the eigenfunctions is provided by Lemma 11.23. Finally, Theorem 11.24 presents an improved error estimate of the eigenvalues by means of the eigenfunctions. The Riesz–Schauder theory also states that the equation \((L-\lambda I)u\;=\;f\)> can even be solved for eigenvalues λ if f satisfies suitable side conditions. These equations are treated in §11.2.4. Section 11.3 discusses the discretisation by difference schemes. Also in this case similar results can be obtained.
Wolfgang Hackbusch

Chapter 12. Stokes Equations

Abstract
Besides differential equations of second or higher order there are systems of q differential equations for q scalar functions. In Section 12.1 we present the systems of the Stokes and Lamé equations as particular examples and define the ellipticity of such systems. Section 12.2 starts with the variational formulation of Stokes’ equations. The saddle-point structure is discussed in §12.2.2. Solvability of general saddle-point problems is analysed in §12.2.3. The corresponding conditions are verified for the Stokes equations. A reinterpretation in §12.2.5 leads to a V 0-elliptic problem in a special subspace V 0. In Section 12.3 the finite-element discretisation is studied. Special inf-sup conditions are to be satisfied since otherwise the problem is not solvable or unstable. Examples of stable finite elements are presented in §12.3.3.
Wolfgang Hackbusch

Backmatter

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