Syllabus: APMA 2560
Introduction to variational calculus – Examples
Theory of the first (Gateaux) variation and the Euler-Lagrange equations
Theory of the second (Gateaux) variation and Legendre’s necessary condition
Variational problems with constraints • Boundary Conditions – Examples
Introduction to the Calculus of Variations, by Hans Sagan (Dover).
Methods of Mathematical Physics, vol. 1, Courant & Hilbert, Chapter IV.
Methods of Weighted Residuals
Galerkin and Petro-Galerkin method
Multi-domain formulation – Finite Elements
Karniadakis and Sherwin, sections 2.1-2.2
Gottlieb and Orszag, section 2
Abstract formulation of FEM for elliptic problems
One dimensional discontinuous FEM
C. Johnson’s book: Chapter 2
Paper by Zhang and Shu
Examples of simple PDEs and their physical properties
Fornberg’s finite difference method for computing derivatives.
Phase error analysis and high-order method; efficiency.
Infinite-order finite difference formula.
B. Fornberg, Generation of high order finite differences on arbitrarily spaced grids, Math. Comput: 51:699-706, 1988.
Fourier Spectral Methods
Fourier-Galerkin method; stability for hyperbolic problems
Fourier-Collocation method; stability for hyperbolic problems
Stability of parabolic equations
Stability of nonlinear equations
Section 6 of Gottlieb and Orszag
Chapter 3 of Hesthaven et al.
Approximation Theory and Resolution Rules-of-Thumb
Chebyshev polynomials and convergence
Regular versus singular Sturm-Liuville problems
Legendre polynomials and convergence
Laguerre and Hermite polynomials
Askey family of orthogonal polynomials
Gottlieb & Orszag, section 3
Karniadakis & Sherwin, Appendix A
Polynomial Spectral Methods
Galerkin formulation; stability
Collocation formulation; stability
Tau and penalty methods
Sections 2-7-8 of Gottlieb & Orszag
Chapters 7-8 of Hesthaven et al.
Spectral Methods for Non-Smooth Problems
Chapter 9 from Hesthaven et al.
Finite Elements in Two-Dimensions: Theory and Implementation
Chapter 4 from C. Johnson
Handout of Lecture 2
Time discretization – Spectral Methods
Stability of continuous and semi-Discrete problems
An example of a fully discrete system (Euler/Legendre collocation)
Eigenspectra of derivative operators – von Neuman stability
Common time-stepping methods (multi-step, multi-stage, integrating factos, and strong-preserving stability)
Read chapter 10 of Hesthaven et al.
Handouts #1 Spectra and eigenspectra (from Canuto et al.)