About this course

MA 5629 is concerned with the analysis and design of algorithms for the numerical solution of partial differential equations.

Announcements

• 11/15: HW #5 posted (pdf)
• 11/15: Bounds for Riesz potentials, interpolation bounds
• 11/10: HW #4 posted (pdf)
• 11/10: remainder terms of Taylor and averaged Taylor polynomials, bounds, chunkiness parameters.
• 11/8: Change of variables in multiple integral, FEM implementation roadmap, Taylor Polynomials (in R^n), cutoff functions, averaged Taylor Polynomials, respective bounds and properties
• 11/3>: Revisiting Hermite cubic elements, concrete example of affine mappings.
• 11/1: Interpolation equivalence, Rectangular elements, Tetrahedrals
• 10/27: Local/Global interpolation, basis functions for trinagular elements, tri_quad.m, affine mapping,
• 10/25: HW #3 posted (pdf)
• 10/25: Triangular finite elements: Lagrange, Hermite, Argyris
• 10/20: Lax-Milgrim, Cea's Lemma, multi-dimensional variational formulation, Finite Element
• 10/18: continuity and coercivity of bilinear form, existence and uniqueness of solution to symmetric variational problems, Contraction mappings
• 10/13: Hilbert spaces, Projections, Riez Representation Theorem
• 10/11: Sobolev norm/inequalities, norms on boundaries, duality
• 10/6: Functional analysis primer, weak derivatives
• 10/4: Weighted norms
• 9/29: local stiffness matrices, pointwise estimates
• 9/27: piecewise quadratic spaces, piecewise_quadratic.pdf
• 9/22: piecewise linear spaces, linear_fe.m
• 9/20: FE Methods: Weak formulation, Ritz-Galerkin approximation, Error Estimates
• 9/15: HW #2 posted (pdf)
• 9/15: FD approximations to the wave equation, overview of Fourier Transforms, dispersion relations, numerical dispersion. In-class code for the linear wave equation: linear_wave.m
• 9/13: Derived and studied the modified equations ofr linear advection (Forward Euler + upwind), Method of Lines, Rothe's Method, Characteristic Tracing.
• 9/8: We revisited FV methods, working through how to solve Burgers' equation using piecewise constant reconstruction and an upwind numerical flux. Then, we returned to Von Neumann stability analysis for linear advection using first-order upwind differencing.
• 9/6: Comparison between FV and FD for Poisson's equation, Lax-equivalence theorem, Von Neumann stabiilty analysis for the heat equation and linear advection. In-class code for linear advection: linear_advection.m
• 9/1: We covered: upwinded finite difference approximations for the advection equation, demonstration that FD converges to the wrong solution for Burgers' equation, formulation of Finite Volume Methods: reconstruction of cell wall values from cell averages, and formulation of numerical fluxes. Useful resources:
  • Chapter 4 in Finite Volume methods for hyperbolic problems by LeVeque, QA377.L41566 2002 (library catalog)
  • Chapter 12 in Numerical Methods for Conservation Laws by LeVeque, QA377.L4157 1992 (library catalog)
  • Chapter 14 in Level Set Methods and Dynamic Implicit Surfaces by Osher and Fekiw, LeVeque, QA1 A647 v.153 (library catalog)
• 8/31: HW #1 posted (pdf)
• 8/30: We covered classifications and examples of common PDEs, finite difference approximation to Poissons's equation in 1D, local truncation error of a difference scheme, order of the difference method, deriving difference formulas using the method of undetermined coefficients and using Lagrange interpolating polynomials, handling Dirichlet and Neumann BC (mirroring technique for the latter)
• 8/29: Syllabus posted (pdf)