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This work presents a thorough treatment of boundary element methods (BEM) for solving strongly elliptic boundary integral equations obtained from boundary reduction of elliptic boundary value problems in $\mathbb{R}^3$. The book is self-contained, the prerequisites on elliptic partial differential and integral equations being presented in Chapters 2 and 3. The main focus is on the development, analysis, and implementation of Galerkin boundary element methods, which is one of the most flexible and robust numerical discretization methods for integral equations. For the efficient realization of the Galerkin BEM, it is essential to replace time-consuming steps in the numerical solution process with fast algorithms. In Chapters 5-9 these methods are developed, analyzed, and formulated in an algorithmic way.

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Preface.- Introduction.- Elliptic Differential Equations.- Elliptic Boundary Integral Equations.- Boundary Element Methods.- Generating the Matrix Coefficients.- Solution of Linear Systems of Equations.- Cluster Methods.- Parametric Surface Approximation.- A Posteriori Error Estimation.- Bibliography.- Index of Symbols.- Index.

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From the reviews:

"The book´s main focus ´is the systematic development of numerical methods to determine the Galerkin solution of boundary integral equations´ (BIEs) in the context of three-dimensional elliptic boundary value problems (BVPs). ... There are separate studies of transmission problems, screen problems, and exterior BVPs for the Helmholtz equation. ... In summary, this is a well-written book on the numerical analysis of Galerkin methods for the solution of boundary integral equations." (Paul Andrew Martin, Mathematical Reviews, Issue 2011 i)

"This book is ... the most comprehensive and self-contained book on BEMs of the ones that exist ... . The book is written rigorously and is an important scholarly contribution to the area of boundary elements. ... this book is an excellent mathematical monograph and valuable reference not only for researchers in the field of BEMs but also for applied mathematicians, engineers, and scientists who are interested in modern computational mathematics. The authors have produced a commendable work of scholarly achievement." (George C. Hsiao, SIAM Review, Vol. 54 (1), March, 2012)

"The focus of the book is the systematic development of numerical methods to determine the Galerkin approximate solution of boundary integral equations ... . The core of the book can be used as the basis for a lecture course on the numerics of boundary integral equation or as an introduction for non-specialists to the development, analysis and implementation of efficient BE methods. Complementary material bridges the gap between a textbook and current research areas, which makes the book useful also for specialists in BEM." (Gunther Schmidt, Zentralblatt MATH, Vol. 1215, 2011)

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