Hromadka, Theodore; Whitley, Robert Foundations of the complex variable boundary element method. (English) Zbl 1295.65094 SpringerBriefs in Applied Sciences and Technology. Cham: Springer (ISBN 978-3-319-05953-2/pbk; 978-3-319-05954-9/ebook). xii, 80 p. (2014). The rough contents of this booklet are as follows: Chapter 1: The heat equation, Chapter 2: Metric spaces, Chapter 3: Banach spaces, Chapter 4: Power series, Chapter 5: The \(\mathbb R^2\) Dirichlet problem, Chapter 6: The \(\mathbb R^N\) Dirichlet problem. The work also contains a short bibliography and a subject index. Each chapter ends with a set of exercises which assists the reader to enhance more accurately the meaning of the theory exposed. The work provides in a clear way the very first fundamentals one needs in numerical approximation of complex variable functions and potential theory. The essentials of Banach spaces functional analysis are reviewed (some of them proven) in order to assure that the complex variable boundary element method can provide reliable approximate solutions to the Dirichlet problem in a bounded domain included in \(\mathbb R^N\), for an arbitrary \(N\) larger than or equal \(2\). Reviewer: Calin Ioan Gheorghiu (Cluj-Napoca) MSC: 65M38 Boundary element methods for initial value and initial-boundary value problems involving PDEs 65-02 Research exposition (monographs, survey articles) pertaining to numerical analysis 65E05 General theory of numerical methods in complex analysis (potential theory, etc.) 30E10 Approximation in the complex plane 65N38 Boundary element methods for boundary value problems involving PDEs 35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation 35K05 Heat equation Keywords:heat equation; Dirichlet problem; Laplace equation; maximum principle; complete normed linear spaces; homogeneous and harmonic polynomials; duality; Hahn-Banach theorem; power series; Weierstrass approximation theorem; Walsh-Lebesgue theorem; Poincaré truncated cone condition; complex variable boundary element method PDFBibTeX XMLCite \textit{T. Hromadka} and \textit{R. Whitley}, Foundations of the complex variable boundary element method. Cham: Springer (2014; Zbl 1295.65094) Full Text: DOI