Solin / Segeth / Dolezel Higher-Order Finite Element Methods

Erscheinungsjahr 2003
ISBN: 978-0-203-48804-1
Verlag: CRC Press
Format: PDF
Kopierschutz: Adobe DRM (»Systemvoraussetzungen)

E-Book, Englisch, 408 Seiten

Reihe: Studies in Advanced Mathematics

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.

Zielgruppe

Applied mathematicians in computation, numerical analysis and partial differential equations; engineers

Autoren/Hrsg.

Solin, Pavel

Segeth, Karel

Dolezel, Ivo

Fachgebiete

• Technische Wissenschaften Maschinenbau | Werkstoffkunde Maschinenbau
• Mathematik | Informatik EDV | Informatik EDV & Informatik Allgemein
• Mathematik | Informatik Mathematik Mathematik Allgemein

Inhaltsverzeichnis

INTRODUCTION

Finite Elements

Orthogonal Polynomials

A One-Dimensional Example

HIERARCHIC MASTER ELEMENTS OF ARBITRARY ORDER

De Rham Diagram H^1-Conforming Approximations

H(curl)-Conforming Approximations

H(div)-Conforming Approximations

L^2-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