A balanced guide to the essential techniques for solving elliptic
partial differential equations
Numerical Analysis of Partial Differential Equations provides a
comprehensive, self-contained treatment of the quantitative methods used
to solve elliptic partial differential equations (PDEs), with a focus on
the efficiency as well as the error of the presented methods. The author
utilizes coverage of theoretical PDEs, along with the nu merical
solution of linear systems and various examples and exercises, to supply
readers with an introduction to the essential concepts in the numerical
analysis of PDEs.
The book presents the three main discretization methods of elliptic
PDEs: finite difference, finite elements, and spectral methods. Each
topic has its own devoted chapters and is discussed alongside additional
key topics, including:
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The mathematical theory of elliptic PDEs
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Numerical linear algebra
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Time-dependent PDEs
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Multigrid and domain decomposition
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PDEs posed on infinite domains
The book concludes with a discussion of the methods for nonlinear
problems, such as Newton's method, and addresses the importance of
hands-on work to facilitate learning. Each chapter concludes with a set
of exercises, including theoretical and programming problems, that
allows readers to test their understanding of the presented theories and
techniques. In addition, the book discusses important nonlinear problems
in many fields of science and engineering, providing information as to
how they can serve as computing projects across various disciplines.
Requiring only a preliminary understanding of analysis, Numerical
Analysis of Partial Differential Equations is suitable for courses on
numerical PDEs at the upper-undergraduate and graduate levels. The book
is also appropriate for students majoring in the mathematical sciences
and engineering.
