Numerical Analysis of Partial Differential Equations

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  MA 8445-6
   Fall 2007/Spring 2008

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• Objective:  This course is intended to be an introduction to the mathematical analysis of finite element (and some finite volume) methods for partial differential equations. The distinctive features of this course are that
  
• it uses the basic concepts of continuous dependence, a priori and posteriori error estimates and duality arguments,
•  deals with the issues of computational complexity and adaptivity, and
  
• emphasizes the interplay between theory and practice.
• References: 
  • D. Braess, Finite Elements. Theory, Fast Solvers and Applications in Solid Mechanics
    
  • F. Brezzi and M. Fortin, Mixed and hybrid finite element methods
    
  • P.G. Ciarlet, The Finite Element Method for Elliptic Problems
    
  • Y. Saad, Iterative Methods for Sparse linear systems
    
  • C. Johnson, Numerical solutions of PDEs by the Finite Element Method
    
  • R. LeVeque, Numerical methods for conservation laws
    
  • R. LeVeque, Finite Volume Methods for Hyperbolic Problems
    
  • A. Quarteroni and A. Valli, Numerical methods for PDEs
    
• Syllabus:
• For the Fall:
  
• The two-point boundary value problem
  
• Examples and their physical interpretation
• The weak formulation and the equivalent energy minimization problem
  
• The continuous Galerkin method: Existence and uniqueness of the approximation
• The matrix equations
  
• Lagrange multipliers and static condensation
  
• A priori error estimates in the energy norm
• Approximation theory
• Algebraic, spectral  and exponential convergence
• A priori error estimates in the L2-norm: The Aubin-Nitsche duality argument
  
• Superconvergence at the nodes for both the potential and its flux
  
• A priori estimates of the error in negative-order norms
• Approximation of linear functionals
  
• Postprocessing by local convolution
• A posteriori estimates in the energy norm
• An adaptive method and its convergence
  
• Second-order elliptic problems in several space dimensions
  
• Examples and their physical interpretation
  
• The general abstract approach
  
• The Lax-Milgam Lemma
  
• The Poincare and trace inequalities: Density and continuity
  
• Finite-dimensional subspaces of  H1: Distributional derivatives
• The continuous Galerkin method
• Definition of a finite element method
  
• Examples of finite element methods
  
• The matrix equations: Implementation details
  
• Lagrange multipliers and static condensation
  
• A priori error estimates in the energy norm
  
• A priori error estimates in the L2-norm: The Aubin-Nitsche duality argument
  
• Elliptic regularity results
  
• A priori error estimates in negative-order norms
• Postprocessing by local convolution
• A posteriori error estimates
  
• An adaptive method and its convergence
  
• Methods for solving the matrix equation
  
• Direct methods
• Gauss elimination
  
• The frontal method
• Classic iterative methods
  
• The Jacobi method
  
• The Gauss-Seidel method
  
• The SSOR method
• Minimization algorithms
  
• The method of steepest descent
  
• Error estimates: The Kantorovich inequality
  
• The method of conjugate gradients
  
• Orthogonality properties
• Error estimates: Chebyshev polynomials
  
• A mixed method
  
• The Raviart-Thomas (RT) method
  
• Existence and uniqueness of the approximation
• The inf-sup condition: Characterization of surjective operators
  
• The commuting-diagram property
• A priori error estimates
  
• Hybridization of the method
  
• Superconvergence of the Lagrance multipliers
  
• Local postprocessing of the approximate solution
• For the Spring:
• Convection-diffusion problems
  
• The model convection-diffusion problem
  
• Continuous Galerkin methos: The need of stabilization
• Streamline-diffusion methods
• Discontinuous Galerkin methods
• The GMRES method
• The conjugate gradient method as a Galerkin method
• Arnoldi's orthogonalization
  
• The full orthogonalization method
• The GMRES method
• Convergence analysis
  
• Convection-diffusion problems
  
• The model convection-diffusion problem
  
• Continuous Galerkin methos: The need of stabilization
• Streamline-diffusion methods
• Discontinuous Galerkin methods
• Linear hyperbolic conservation laws
  
• The transport equation
  
• The discontinuous Galerkin space discretization
• A priorir error analysis of the semidiscrete method
  
• The RKDG method
  
• Stability and accuracy of the method
  
• Amplitude and phase errors
• Friedrich's systems
  
• Non-linear scalar hyperbolic conservation laws
  
• The loss of well-posedness
  
• Traveling wave solutions: The entropy condition
• The Riemann problem
• The entropy solution
  
• Continuous dependence results
• Monotone schemes
• Convergence and error estimates
• The RKDG method
• Compressible fluid flow
  
• The Euler equations for gas dynamics
  
• The RKDG method 
  
• The compressible Navier-Stokes equations
  
• The RKDG method
  
•  Incompressible fluid flow
• The Stokes, Oseen and incompressible Navier-Stokes equations
  
• Mixed methods for the Stokes equations
  
• Discontinuous Galerkin methods for the Stokes equations
• Obtaining exactly incompressible velocity approximations
  
• Hybridization
  
• Discontinuous Galerkin methods for the Oseen equations
  
• Discontinuous Galerkin methods for the Navier-Stokes equations