Higher-Order Finite Element Methods

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.

Contents

• 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