Numerical Approximation of Partial Differential Equations

verfasst von: Sören Bartels

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

Finite element methods for approximating partial differential equations have reached a high degree of maturity, and are an indispensible tool in science and technology. This textbook aims at providing a thorough introduction to the construction, analysis, and implementation of finite element methods for model problems arising in continuum mechanics. The first part of the book discusses elementary properties of linear partial differential equations along with their basic numerical approximation, the functional-analytical framework for rigorously establishing existence of solutions, and the construction and analysis of basic finite element methods. The second part is devoted to the optimal adaptive approximation of singularities and the fast iterative solution of linear systems of equations arising from finite element discretizations. In the third part, the mathematical framework for analyzing and discretizing saddle-point problems is formulated, corresponding finte element methods are analyzed, and particular applications including incompressible elasticity, thin elastic objects, electromagnetism, and fluid mechanics are addressed. The book includes theoretical problems and practical projects for all chapters, and an introduction to the implementation of finite element methods.

Inhaltsverzeichnis

Frontmatter

Finite Differences and Finite Elements

Frontmatter

Chapter 1. Finite Difference Method

Abstract

The chapter discusses the mathematical description of transport, diffusion, and wave phenomena and their numerical simulation with finite difference methods. The accuracy of the methods is investigated via stability and consistency properties assuming the existence of regular solutions. Optimal order convergence rates for general boundary conditions are addressed. The practicality of the methods is illustrated with short implementations.

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Chapter 2. Elliptic Partial Differential Equations

Abstract

General existence theories for solutions of partial differential equations require using concepts from functional analysis and considering generalizations of classical derivatives based on a multidimensional integration-by-parts formula. The chapter introduces Sobolev spaces, discusses their main properties, states existence theories for elliptic second order linear partial differential equations, and sketches regularity results for solutions.

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Chapter 3. Finite Element Method

Abstract

Finite element methods provide an abstract framework for interpolating functions or vector fields in multidimensional domains. They allow for specifying Galerkin methods for approximating partial differential equations. In combination with regularity results, error estimates in various norms can be proved. The efficient implementation of low order and isoparametric methods is discussed in the case of stationary and evolutionary model problems.

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Local Resolution and Iterative Solution

Frontmatter

Chapter 4. Local Resolution Techniques

Abstract

Convergence rates of standard numerical methods are suboptimal when the solution has corner singularities as in the case of elliptic equations on nonconvex domains. Optimal rates can be obtained using locally refined triangulations which are either specifically constructed for particular domains or generated automatically via adaptive mesh-refinement algorithms. Both approaches are introduced, their convergence is analyzed, and their implementation is illustrated.

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Chapter 5. Iterative Solution Methods

Abstract

Linear systems of equations resulting from finite element discretizations of partial differential equations are typically large, sparse, and ill-conditioned. Their efficient numerical solution exploits properties of the underlying continuous problem or a sequence of discretizations. The chapter discusses multigrid, domain decomposition, and preconditioning methods.

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Constrained and Singularly Perturbed Problems

Frontmatter

Chapter 6. Saddle-Point Problems

Abstract

Standard numerical methods fail to provide accurate approximations when partial differential equations involve constraints defined by a differential operator or when they contain terms weighted by a larger parameter. Generalizations of the Lax–Milgram and Céa lemmas provide a concise framework for the development and analysis of appropriate numerical methods. Central to their construction is the validity of an inf-sup condition that defines a compatibility requirement on involved finite element spaces.

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Chapter 7. Mixed and Nonstandard Methods

Abstract

Stable finite element methods for discretizing a saddle-point formulation of the Poisson problem and the Stokes system are introduced and analyzed. Characteristic properties of convection-dominated equations and their numerical approximation via introducing stabilizing terms is investigated. Flexible discontinuous Galerkin methods are derived and analyzed for a model Poisson equation. Simple implementations illustrate the performance of the numerical methods.

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Chapter 8. Applications

Abstract

The chapter discusses the development, analysis, and implementation of numerical methods for boundary value problems in elasticity, electromagnetism, and fluid mechanics. Each of the considered problems requires a suitable numerical treatment to capture relevant effects with a low number of degrees of freedom.

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Backmatter

Titel: Numerical Approximation of Partial Differential Equations
verfasst von: Sören Bartels
Verlag: Springer International Publishing
Electronic ISBN: 978-3-319-32354-1
Print ISBN: 978-3-319-32353-4
DOI: https://doi.org/10.1007/978-3-319-32354-1