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Polynomial Approximation of Differential Equations,

Lecture Notes in Physics, Volume 8,

Springer-Verlag, Heidelberg 1992, p. X+303

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1. Special families of polynomials

  • Sturm-Liouville problems
  • The Gamma function
  • Jacobi polynomials
  • Legendre polynomials
  • Chebyshev polynomials
  • Laguerre polynomials
  • Hermite polynomials

2. Orthogonality

  • Inner products and norms
  • Orthogonal functions
  • Fourier coefficients
  • The projection operator
  • The maximum norm
  • Basis transformations

3. Numerical integration

  • Zeroes of orthogonal polynomials
  • Lagrange polynomials
  • The interpolation operators
  • Gauss integration formulas
  • Gauss-Lobatto integration formulas
  • Gauss-Radau integration formulas
  • Clenshaw-Curtis integration formulas
  • Discrete norms
  • Discrete maximum norms
  • Scaled weights

4. Transforms

  • Fourier transforms
  • Aliasing
  • Fast Fourier transform
  • Other fast methods

5. Functional spaces

  • The Lebesgue integral
  • Spaces of measurable functions
  • Completeness
  • Weak derivatives
  • Transformation of measurable functions in R
  • Sobolev spaces in R
  • Sobolev spaces in intervals

6. Results in approximation theory

  • The problem of best approximation
  • Estimates for the projection operator
  • Inverse inequalities
  • Other projection operators
  • Convergence of the Gaussian formulas
  • Estimates for the interpolation operators
  • Laguerre and Hermite functions
  • Refinements

7. Derivative Matrices

  • Derivative matrices in the frequency space
  • Derivative matrices in the physical space
  • Boundary conditions in the frequency space
  • Boundary conditions in the physical space
  • Derivatives of scaled functions
  • Numerical solution

8. Eigenvalue analysis

  • Eigenvalues of first derivative operators
  • Eigenvalues of higher-order operators
  • Condition number
  • Preconditioners for second-order operators
  • Preconditioners for first-order operators
  • Convergence of eigenvalues
  • Multigrid method

9. Ordinary differential equations

  • General considerations
  • Approximation of linear equations
  • The weak formulation
  • Approximation of problems in the weak form
  • Approximation in unbounded domains
  • Other techniques
  • Boundary layers
  • Nonlinear equations
  • Systems of equations
  • Integral equations

10. Time-dependent problems

  • The Gronwall inequality
  • Approximation of the heat equation
  • Approximation of linear first-order problems
  • Nonlinear time-dependent problems
  • Approximation of the wave equation
  • Time discretization

11. Domain-decomposition methods

  • Introductory remarks
  • Non-overlapping multidomain methods
  • Solution techniques
  • Overlapping multidomain methods

12. Examples

  • A free boundary problem
  • An example in an unbounded domain
  • The nonlinear Schroedinger equation
  • Zeroes of Bessel functions

13. An example in two dimensions

  • Poisson’s equation
  • Approximation by the collocation method
  • Hints for the implementation
  • The incompressible Navier-Stokes equations



Spectral Elements for Transport-Dominated Equations,

Lecture Notes In Computational Science and Engineering, Volume 1,

Springer-Verlag, New York 1997, p. X+212

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1. The Poisson equation in the square

  • Statement of the problem and preliminary results
  • The collocation method for the Poisson equation
  • Convergence analysis of the collocation method
  • Numerical implementation of the collocation method
  • The numerical algorithm for the Poisson equation
  • Application to the eikonal equation
  • More about boundary conditions
  • A Sturm-Liouville problem

2. Steady transport-diffusion equations

  • A more general boundary-value problem
  • The upwind grid
  • The collocation method at the upwind grid
  • The numerical algorithm for the transport-diffusion equation

3. Other kinds of boundary conditions

  • Neumann-type boundary conditions
  • Boundary conditions in weak form
  • The numerical algorithm for the Neumann problem
  • More general boundary conditions
  • An approximation of the Poincar\’e-Steklov operator

4. The spectral element method

  • Complex geometries
  • The Poisson equation in a complex domain
  • Approximation by spectral elements
  • An iterative algorithm for the domain decomposition method
  • Improvements
  • Transport-diffusion equations in complex geometries <\LI>
  • A model problem in Electrostatics

5. Time discretization

  • Time-dependent advection-diffusion problems
  • The incompressible Navier-Stokes equations
  • Approximation of the Navier-Stokes equations
  • The nonlinear Schroedinger equation
  • Semiconductor device equations

6. Extensions

  • A posteriori error estimators
  • Pure hyperbolic problems
  • Elements with bent sides: application to the dam problem
  • The Poisson equation in 3-D

A. Appendix

  • Characterizing properties of Legendre polynomials
  • Zeroes and quadrature formulas
  • Interpolation and evaluation of derivatives
  • Approximation results


Electromagnetism and the Structure of Matter,

 World-Scientific, Singapore, 2008, xii+190

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 book review  of  MathSciNet


1. Something is wrong with classical electromagnetism

  • Maxwell equations and wave-fronts
  • Wave-front propagation
  • Fronts from an oscillating dipole
  • Preliminary conclusions

2. First steps towards the new model

  • Modified Maxwell equations
  • Perfect spherical waves
  • Travelling signal packets
  • Lagrangian formulation
  • Free-waves and the eikonal equation
  • Lorentz invariance

3. Interaction of waves with matter

  • Waves bouncing off an obstacle
  • Diffraction phenomena
  • Adding the mechanical terms
  • Properties of the new set of equations

4. The equations in the framework of general relativity

  • Preliminary considerations
  • The energy tensor
  • Unified field equations
  • The divergence of the magnetic field

5. Building matter from fields

  • Adding the pressure tensor
  • On the existence of particle-like solutions
  • Looking for 2-D constrained waves
  • Neutrinos, electrons and protons
  • Connections with a Dirac type equation

6. Final speculative considerations

  • Towards deterministic quantum mechanics
  • Conclusions