Course Topics for CS 558 / CSE 513 : Calculus on Meshes

List of Topics:
  1. Introduction
    • When do traditional vertex based finite element methods fail ?
    • Solving PDEs, finding handles and holes in meshes, vector field decomposition.
    • Simplicial complexes, chain and cochain complexes.
    • Introduction to differential forms and vector fields.
    • Exterior derivative, wedge product, Hodge star, sharps and flats, codifferential, Laplacian.
    • Inner product of differential forms.

  2. Discrete Exterior Calculus
    • Discrete differential forms as cochains.
    • Discrete exterior derivative as coboundary.
    • Well-centered meshes and circumcentric dual meshes.
    • Diagonal inner product of discrete differential forms.
    • Discrete Hodge star and codifferential for well-centered meshes.
    • Discrete Laplacian for well-centered meshes.
    • Flow in porous media (Darcy flow).
    • Finding holes in planar triangle meshes.

  3. Mixed Finite Element Methods
    • Introduction to finite element methods.
    • Function spaces H(div) and H(curl).
    • Introduction to mixed finite element methods.
    • The inf-sup or LBB condition for stability of mixed methods.
    • Element zoo : Raviart-Thomas, Nedelec, Edge and Face elements.

  4. Finite Element Exterior Calculus
    • Interpreting Discrete Exterior Calculus as a finite element method.
    • Interpreting and generalizing mixed FEM in terms of exterior calculus.
    • Whitney map and Whitney forms and their generalization.
    • Whitney inner product of discrete differential forms.
    • Discrete Hodge star and codifferential for arbitrary meshes.
    • Discrete Laplacian for arbitrary meshes.
    • Resonant electromagnetic cavity (curl-curl) problem.

  5. Discrete Hodge Theory
    • User's guide to homology and de Rham cohomology.
    • Hodge decomposition of discrete differential forms.
    • Discrete vector field decomposition into divergence-free, curl-free and harmonic parts.
    • Finding handles and holes in surface meshes and other complexes.
    • Computation of discrete harmonic forms.
Page maintained by Anil N. Hirani and Evan VanderZee. Last updated October 3, 2008.