3R1. Finite Element Methods with B-Splines. - Edited by K Hollig (Universitat Stuttgart, Stuttgart, Germany). SIAM, Philadelphia. 2003. 145 pp. ISBN 0-89871-533-4. $65.00.
Reviewed by DJ Benson (Dept of Appl Mech and Eng Sci, UCSD, 9500 Gilman, La Jolla CA 92093-0411).
The finite element method has been used extensively for solving partial differential equations for over forty years. During most of this period, the basis functions used to approximate the solution have primarily been piecewise continuous polynomials typically derived from Lagrange interpolating polynomials. Recently, other classes of basis functions have been explored, for example, the element free Galerkin methods. This slim book (145 pages, including the index and references) describes the use of B-splines as the basis functions in the finite element method.
Beyond the choice of basis functions, the formulation described in this book departs in two important aspects from what may be regarded as the standard finite element formulation found in textbooks.
First, the formulation uses a logically regular mesh instead of an unstructured mesh. Second, the boundaries of the domain run through the elements and don’t correspond to mesh lines. The essential boundary conditions are therefore not directly enforced by specifying nodal values, but by the use of web-splines, which are splines modified by weight functions.
This book is not a textbook, but a monograph on current research. It appears, in essence, to be a collection of the eight publications by Hollig referenced in the text. Since not all of the publications are in journals, or in English, the publication is valuable to readers in the United States who are interested in the author’s approach. The target audience is clearly the finite element researcher. Since the formulation isn’t in any commercial codes, those seeking to understand how typical finite element codes work should look elsewhere. Compared to most engineering finite element textbooks, the mathematical level is high. However, if the reader is comfortable with the material in an introductory book on the mathematics of the finite element method such as the classic by Strang and Fix, the mathematics in the book will not be a challenge. Graduate students interested in the finite element method, but not as their research topic, may find the book disappointing. Others, however, may appreciate it for its original ideas and nontraditional approach.
The ideas presented in Finite Element Methods with B-Splines are very interesting, however, given the novelty of the material, its presentation is overly terse even for a monograph. The author provides a solid list of references, however he relies on them too heavily instead of presenting their ideas in detail. The second chapter (16 pages) covers basic finite element concepts, including Sobolev spaces, abstract variational problems, and approximation errors. The third chapter covers B-splines; the fourth, their application as a finite element basis; and the fifth, a mathematical analysis of the formulation. Chapter 6 describes the application of the formulation, in a little over twenty pages, to Poisson’s equation, mixed problems with variable coefficients, the biharmonic equation, and linear elasticity, while multigrid methods are covered in less than twenty pages in Chapter 7. The last chapter describes implementation issues at a level of abstraction that will not help the novice and is trivial to experienced developers.