Higher-Order Finite Element Methods

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Overview

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 H1, 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.

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Product Details

Table of Contents

INTRODUCTION
Finite Elements
Orthogonal Polynomials
A One-Dimensional Example
HIERARCHIC MASTER ELEMENTS OF ARBITRARY ORDER
De Rham Diagram H1-Conforming Approximations
H(curl)-Conforming Approximations
H(div)-Conforming Approximations
L2-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
Constrained Approximation
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
Goal-Oriented Adaptivity
Automatic Goal-Oriented h-, p-, and hp-Adaptivity
Automatic Goal-Oriented hp-Adaptivity in Two Dimensions

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