Contents

• Contents
• List of Tables
• List of Figures
• Introduction
  • Objectives
  • Outline
• The Elements
  • The Triangle
  • The Quadrilateral
  • The Tetrahedron
  • The Pyramid
  • The Prism
  • The Hexahedron
  • Global Assembly
    • Algorithm 1
    • Algorithm 2
    • Algorithm For Connecting Prisms And Tetrahedra
• Basis Functions
  • One-Dimensional Orthogonal Polynomials
  • Singular Sturm-Liouville Problems on a Simplex in d-Dimensions
  • Orthogonal (Modal) Bases
    • The Orthogonal Triangle Basis
    • The Orthogonal Quadrilateral Basis
    • The Orthogonal Prism Basis
    • The Orthogonal Hexahedral Basis
    • The Decay of Basis Coefficients
  • Semi-Orthogonal Bases
    • Modal basis
      • Triangle Basis: ${\bf TrHH}$
      • Quadrilateral Bases: ${\bf QuHH}$ and ${\bf QuHN}$
    • Nodal and Mixed Bases
      • Quadrilateral Basis: ${\bf QuNN}$
      • Triangle Basis: ${\bf TrNH}$
  • Comparison of the Bases
• Operators
  • Elemental Spatial Operators
    • Integration
    • Flux Integrals
    • Differentiation
    • Implementation and Programming
  • Global Projection Operator
    • Global Convergence in Skew Elements
  • Convective Operator
    • Accuracy of the Convective Operator
    • Spectrum of the Galerkin 2D Convective Operator
    • Three-Dimensional Galerkin Convective Operator Spectrums
    • Spectrum of the Galerkin 3D Convective Operator
  • Diffusion Operator
    • The Schur Complement Method of Inverting Global Operators
    • Optimal Numbering for the Schur Complement Boundary Matrix
    • Numerical Results
    • Convergence for the Galerkin Helmholtz Operator
      • Convergence in 3D Hybrid Elements
      • Convergence in 3D Skew Elements
• Incompressible Flow Simulations
  • Incompressible Navier-Stokes Equation
    • Formulation
    • Summary of Scheme, Boundary Conditions and Implementation
    • Wannier Flow
    • Kovasznay Flow
    • Cylinder Flow
    • The Robotuna
  • ALE Incompressible Navier-Stokes
    • Formulation
    • Temporal Discretization
    • Rayleigh Sliding Plate Problem
    • Design of a Micro-Pump via Simulation
    • Micro-Pump Geometric Specifications
    • Efficiency Analysis
    • Numerical Simulation
• Incompressible Viscous Magnetohydrodynamics
  • Formulation
  • Numerical Scheme
  • Magnetic Pearson's Vortex
  • Orszag-Tang Vortex
  • Cylinder Flow
• Compressible Flow Simulations
  • Numerical Formulation
    • Discontinuous Galerkin for Diffusion
  • Compressible Navier-Stokes Simulations
    • Convergence
    • Cylinder Flow
    • Flow Past a Two-Dimensional NACA0012 Airfoil
    • Flow Past a Multi-Body Airfoil
    • Flow Past A NACA0012 Wing With Endplates
• Compressible Viscous Magnetohydrodynamics Simulations
  • Formulation
  • The $\nabla \cdot {\bf B}=0$ Constraint
  • Implementation of the Euler Flux Terms
  • Implementation of the Viscous Terms
  • Summary of the Compressible MHD Code Implementation
  • Two-Dimensional Magnetohydrostatic Example
  • Three-Dimensional Magnetohydrostatic Example
  • Compressible Orszag-Tang Vortex
  • Flow Past a Cylinder
• Summary
• Continuous Expansion Basis
  • Jacobi Polynomials
  • Modal Basis
    • Two-Dimensional Expansions
      • Quadrilateral Domain
      • Triangular Domain
    • Three-Dimensional Expansions
      • Hexahedral Domain
      • Prismatic Domain
      • Pyramidic Domain
      • Tetrahedral Domain
  • Nodal Basis
    • Two-Dimensional Expansions
      • Quadrilateral Domain
    • Three-Dimensional Expansions
      • Hexahedral Domain
• References