Higher-Order Finite Element Methods
This book is concerned with the treatment of basic principles of higher-order finite element techniques. It consists of six chapters; the first is an introduction, giving a survey of function spaces, defining finite elements, orthogonal polynomials and presenting a one-dimensional example. Chapter 2 is devoted to hierarchic higher-order master elements in 2D and 3D. This chapter contains a number of formulae of shape functions for H1-, H(curl)- and H(div)-conforming elements. Chapter 3 explains basic principles of general higher-order finite element methods in 2D and 3D, namely projection-based interpolation, transfinited interpolation, construction of reference maps, technology of discretization in 2D and 3D and constrained approximations including the case of hanging nodes discretization. Moreover some software aspects are discussed. Chapter 4 deals with higher-order numerical integration for various types of finite elements in two and three space dimensions, e.g. Newton-Cotes, Chebyshev, Lobatto (Radau) and Gauss quadratures and formulae for reference triangle, quadrilateral, brick, tetrahedron and prism. Chapter 5 discusses numerical algorithms for the solution of the finite element discrete problem: direct methods (Gauss elimination, matrix factorization) and iterative methods (steepest decent, ORTHOMIN, conjugate gradients, MINRES, GMRES, preconditioning, block iterative methods, multigrid) and methods for the solution of large systems of ordinary differential equations. Finally, in Chapter 6, mesh optimization, reference solutions and hp-adaptivity are presented. Here, various adaptive strategies and goal adaptivity approaches are treated. The book represents an excellent and useful introduction to higher-order finite element techniques. It is well written and contains a lot of important material. It will satisfy the interest of applied mathematicians as well as engineers. The book is also suitable for advanced undergraduate, graduate and postgraduate students of mathematics and technical sciences.