The boundary-domain integral method for elliptic systems / Andreas Pomp

Beside the finite element method (FEM), the integral equation methods meet an increasing success to approximate the solutions of partial differential equations. A first kind of integral equation methods is given by the boundary element methods (BEM). Since they require the knowledge of a fundamental solution, their use in the approximation of general thin shell equations is quite difficult, or even impossible.

This book studies a nice alternative, named “boundary-domain integral method” (BDIM) which couples boundary and domain integral equations and which can handle variable coefficients like in general thin shell equations. The aim of this book is to give a strong mathematical foundation of the BDIM from a numerical point of view and a detailed description of the complete algorithm. The contents can be summarized as follows:

Part I (3 chapters) is dedicated to the construction of Levi functions of arbitrary degree of elliptic systems of partial differential equations with variable coefficients.

Part II (4 chapters) considers the application of the general theory to the Donnell-Vlasov shell model. This model includes shells with a smooth and arbitrary curved mid-surface, the boundary of which may have corners.

From this book, three main results can be retained: i) the algorithm to construct Levi functions; ii) the analysis of the order of error estimates (Theorem 6.10.1); iii) the discussion about the numerical effectiveness of BDIM and the comparison with FEM.

This short monograph (163 pages) is attractive and nicely written and should be a brilliant alternative to more classical BEM approaches. Potential interested readers are applied mathematicians (mainly) and computational mechanicians at PhD level. (Zentralblatt)