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Applied Functional Analysis, Third Edition provides a solid
mathematical foundation for the subject. It motivates students to
study functional analysis by providing many contemporary
applications and examples drawn from mechanics and science. This
well-received textbook starts with a thorough introduction to
modern mathematics before continuing with detailed coverage of
linear algebra, Lebesque measure and integration theory, plus
topology with metric spaces. The final two chapters provides
readers with an in-depth look at the theory of Banach and Hilbert
spaces before concluding with a brief introduction to Spectral
Theory. The Third Edition is more accessible and promotes interest
and motivation among students to prepare them for studying the
mathematical aspects of numerical analysis and the mathematical
theory of finite elements.
With a focus on 1D and 2D problems, the first volume of Computing
with hp-ADAPTIVE FINITE ELEMENTS prepared readers for the concepts
and logic governing 3D code and implementation. Taking the next
step in hp technology, Volume II Frontiers: Three-Dimensional
Elliptic and Maxwell Problems with Applications presents the
theoretical foundations of the 3D hp algorithm and provides
numerical results using the 3Dhp code developed by the authors and
their colleagues. The first part of the book focuses on
fundamentals of the 3D theory of hp methods as well as issues that
arise when the code is implemented. After a review of
boundary-value problems, the book examines exact hp sequences,
projection-based interpolation, and De Rham diagrams. It also
presents the 3D version of the automatic hp-adaptivity package, a
two-grid solver for highly anisotropic hp meshes and goal-oriented
Krylov iterations, and a parallel implementation of the 3D code.
The second part explores several recent projects in which the 3Dhp
code was used and illustrates how these applications have greatly
driven the development of 3D hp technology. It encompasses acoustic
and electromagnetic (EM) scattering problems, an analysis of
complex structures with thin-walled components, and challenging
simulations of logging tools. The book concludes with a look at the
future of hp methods. Spearheaded by a key developer of this
technology with more than 20 years of research in the field, this
self-contained, comprehensive resource will help readers overcome
the difficulties in coding hp-adaptive elements.
Offering the only existing finite element (FE) codes for Maxwell
equations that support hp refinements on irregular meshes,
Computing with hp-ADAPTIVE FINITE ELEMENTS: Volume 1. One- and
Two-Dimensional Elliptic and Maxwell Problems presents 1D and 2D
codes and automatic hp adaptivity. This self-contained source
discusses the theory and implementation of hp-adaptive FE methods,
focusing on projection-based interpolation and the corresponding
hp-adaptive strategy. The book is split into three parts,
progressing from simple to more advanced problems. Part I examines
the hp elements for the standard 1D model elliptic problem. The
author develops the variational formulation and explains the
construction of FE basis functions. The book then introduces the 1D
code (1Dhp) and automatic hp adaptivity. This first part ends with
a study of a 1D wave propagation problem. In Part II, the book
proceeds to 2D elliptic problems, discussing two model problems
that are slightly beyond standard-level examples: 3D axisymmetric
antenna problem for Maxwell equations (example of a complex-valued,
indefinite problem) and 2D elasticity (example of an elliptic
system). The author concludes with a presentation on infinite
elements - one of the possible tools to solve exterior
boundary-value problems. Part III focuses on 2D time-harmonic
Maxwell equations. The book explains the construction of the hp
edge elements and the fundamental de Rham diagram for the whole
family of hp discretizations. Next, it explores the differences
between the elliptic and Maxwell versions of the 2D code, including
automatic hp adaptivity. Finally, the book presents 2D exterior
(radiation and scattering) problems and sample solutionsusing
coupled hp finite/infinite elements. In Computing with hp-ADAPTIVE
FINITE ELEMENTS, the information provided, including many
unpublished details, aids in solving elliptic and Maxwell problems.
Applied Functional Analysis, Third Edition provides a solid
mathematical foundation for the subject. It motivates students to
study functional analysis by providing many contemporary
applications and examples drawn from mechanics and science. This
well-received textbook starts with a thorough introduction to
modern mathematics before continuing with detailed coverage of
linear algebra, Lebesque measure and integration theory, plus
topology with metric spaces. The final two chapters provides
readers with an in-depth look at the theory of Banach and Hilbert
spaces before concluding with a brief introduction to Spectral
Theory. The Third Edition is more accessible and promotes interest
and motivation among students to prepare them for studying the
mathematical aspects of numerical analysis and the mathematical
theory of finite elements.
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