The finite element method has always been a mainstay for solving engineering problems numerically. The most recent developments in the field clearly indicate that its future lies in higher-order methods, particularly in higher-order hp-adaptive schemes. These techniques respond well to the increasing complexity of engineering simulations and satisfy the overall trend of simultaneous resolution of phenomena with multiple scales.
Higher-Order Finite Element Methods provides an thorough survey of intrinsic techniques and the practical know-how needed to implement higher-order finite element schemes. It presents the basic priniciples of higher-order finite element methods and the technology of conforming discretizations based on hierarchic elements in spaces H^1, H(curl) and H(div). The final chapter provides an example of an efficient and robust strategy for automatic goal-oriented hp-adaptivity.
Although it will still take some time for fully automatic hp-adaptive finite element methods to become standard engineering tools, their advantages are clear. In straightforward prose that avoids mathematical jargon whenever possible, this book paves the way for fully realizing the potential of these techniques and putting them at the disposal of practicing engineers.
“[A] large number of constructions on hp finite elements … are collected and presented in a unified and consistent way. In this respect the book is timely, very useful, and quite unique. … [The book] contains many detailed constructions, interesting concepts, and practical discussions. … It could be used as a reference book and/or a supplement to a course on discretization methods for differential equations.”
— Mathematics of Computation, April 2005
“This book is a valuable addition to the existing literature on adaptive hp-finite element methods [1, 2, 3] and provides the practitioner with the necessary tools and techniques for understanding and implementing high-order hierarchic finite element methods. … Overall this is a useful and well-written text. … I recommend the book for applied mathematicians and practitioners using the finite element method.”
— SIAM Review
“This book gives a very thorough explanation of how to construct an hp-finite-element computer program.”
"… a timely and balanced introduction to the very active research area of hp-adaptive methods for the solution of partial differential equations. … The detailed overview of element types and quadrature rules provides a valuable reference, while the discussion of suitable data structures and efficient linear solvers provide an excellent introduction to key aspects, often neglected, of this class of methods. … a delightful introduction to these rapidly expanding research areas."
— Jan S. Hesthaven, Division of Applied Mathematics, Brown University
"… the overall structure of this comprehensive monograph is very transparent and the style of presentation of the material has an expository nature. … There is no doubt that this new monograph on recent topics of finite element theory and applications will be welcomed by the broad community of finite element researchers and lecturers, both by the practitioners and by the numerical functional analysts."
— Roger Van Keer, Ghent University, Belgium
"….Summing up, the book is well-written, and the overall structure is clear and well-explained. I think that it surely will meet the interest of applied mathematician, engineers, and practitioners."
—Alberto Valli, Universita Degli Studi Di Trento, Italy
A One-Dimensional Example
HIERARCHIC MASTER ELEMENTS OF ARBITRARY ORDER
De Rham Diagram H^1-Conforming Approximations
HIGHER-ORDER FINITE ELEMENT DISCRETIZATION
Projection-Based Interpolation on Reference Domains
Transfinite Interpolation Revisited
Construction of Reference Maps
Projection-Based Interpolation on Physical Mesh Elements
Technology of Discretization in Two and Three Dimensions
Selected Software-Technical Aspects
HIGHER-ORDER NUMERICAL QUADRATURE
One-Dimensional Reference Domain K(a)
Reference Quadrilateral K(q)
Reference Triangle K(t)
Reference Brick K(B)
Reference Tetrahedron K(T)
Reference Prism K(P)
NUMERICAL SOLUTION OF FINITE ELEMENT EQUATIONS
Direct Methods for Linear Algebraic Equations
Iterative Methods for Linear Algebraic Equations
Choice of the Method
Solving Initial Value Problems for ordinary Differential Equations
MESH OPTIMIZATION, REFERENCE SOLUTIONS, AND hp-ADAPTIVITY
Automatic Mesh Optimization in One Dimension
Adaptive Strategies Based on Automatic Mesh Optimization
Automatic Goal-Oriented h-, p-, and hp-Adaptivity
Automatic Goal-Oriented hp-Adaptivity in Two Dimensions