Class Information

Course Description

Class Information

  • Hours: Mondays and Wednesdays 11:40-12:55 pm CT
  • Location: UTK:  Doherty 406                    UTSI: Main Academic Building E110

Course notes from ME517 (CFEM)

Assignments

  • HW1: Verify if there is any way to make the stiffness matrix for DG’s thermal formulation symmetric (Due 2/5/2018).
  • HW2: link  DG/CFEM; Explicit/Implicit (Due 2/19/2018).
  • Matlab and config files for 1D elliptic, parabolic, hyperbolic PDEs (DG formulation) can be downloaded from http://rezaabedi.com/wp-content/uploads/Courses/DG/DG_1D_PDEs.zip
    • HW3: (link) Elliptic PDEs (this includes problem description for all PDE types and two part assignment for elliptic PDEs).
      • Part 1, due 2/21/2018
      • Part 2: due 2/26/2018
    • HW4: (link) Parabolic PDEs
      • Part 1, due 3/5/2018
      • Part 2: due 3/12/2018
    • HW5: (link) Hyperbolic PDEs
      • Part 1, due 3/19/2018
      • Part 2: due 3/26/2018
    • HW6: (link) SDG method, Due 4/9/2018
    • HW7: (link) Riemann solutions, Due 4/18/2018

Reading Assignments

  • DG vs CFEM:
    • Reference [2]: Introduction (pages 1 to 10), particularly the discussion around Table 1.1
    • Reference [3]: Section 2.4
    • Reference [4]: Pages 64-67.
  • Arnold (2000): sections 1 and 2, table on page 11.
  • Arnold (2002): Up to and including section 3.2 (or 3.6), especially Tables 3.1, Fig. 3.1, Table 6.1, also see very convenient definition of jump and average operators for vector q and scalar  at the bottom of page 7 and top of page 8.
  • Penalty methods versus fluxed based DG formulations:
    • Reference [4]: Pages 3-6.
    • Regarding unification of penalty methods and interior penalty methods, refer to reading assignments for Arnold (2000) and Arnold (2002).
  • Fluxes for parabolic PDEs based on a local solution to an interface problem (with different diffusion coefficients on the two sides of the interface): Lorcher, et. al. An explicit discontinuous Galerkin scheme with local time-stepping for general unsteady diffusion equations, 2008, especially Appendix A.

 

Class timeline

Hand-written notes: pdf                   onenote

  1. 01/22/2018          Lecture: notes      Comparison of continuous and discontinuous Galerkin FEMs. Sample thermal problem: CFEM formulation.
  2. 01/24/2018          Lecture: notes      Comparison of continuous and discontinuous Galerkin FEMs. Sample thermal problem: CFEM formulation/ DG formulation (part 1).
  3. 01/29/2018          Lecture: notes      Sample thermal problem: DG formulation (part 2).
  4. 01/31/2018          Lecture: notes       Sample thermal problem: DG formulation (part 3).
  5. 02/05/2018          Lecture: notes     Explicit versus Implicit time integration for a parabolic PDE / DG vs. CFEM.
  6. 02/07/2018          Lecture: notes     Explicit versus Implicit (part 2), DG fluxes and relation to interior penalty methods (part 1).
  7. 02/12/2018          Lecture: notes      DG fluxes and relation to interior penalty methods (part 2).
  8. 02/14/2018          Lecture: notes      DG fluxes and relation to interior penalty methods (part 3).
  9. 02/19/2018          Lecture: notes      DG fluxes and relation to interior penalty methods (part 4).
  10. 02/21/2018          Lecture: notes      DG formulation for hyperbolic PDEs (part 1).
  11. 02/26/2018          Lecture: notes      DG formulation for hyperbolic PDEs (part 2).
  12. 02/28/2018          Lecture: notes      DG formulation for hyperbolic PDEs (part 3); Spacetime methods: expression of balance laws in spacetime (part 1).
  13. 03/05/2018          Lecture: notes      Spacetime methods: expression of balance laws in spacetime (part 2).
  14. 03/05/2018          Lecture: notes      Spacetime methods: Weighted residual and weak forms.
  15. 03/19/2018          Lecture: notes       Spacetime methods: Weighted residual and weak forms for a hyperbolic heat equation.
  16. 03/21/2018          Lecture: notes       asynchronous Spacetime Discontinous Galerkin (aSDG) method (part 1).
  17. 03/26/2018          Lecture: notes       aSDG method (part 2). Method of characteristics for the solution of systems of conservation laws (1)
  18. 03/28/2018          Lecture: notes       characteristics / solution of systems of conservation laws (2); Riemann solutions: time-marching DG methods.
  19. 04/02/2018          Lecture: notes       Riemann solutions: time-marching SDG methods; Riemann solutions for material interfaces (part 1).
  20. 04/04/2018          Lecture: notes       Riemann solutions for material interfaces (part 2) [method of characteristics] .
  21. 04/06/2018          Lecture: notes       Riemann solutions for material interfaces (part 3); [ use of jump conditions]; side note: scattering coefficients.
  22. 04/09/2018          Lecture: notes       Riemann solutions for material interfaces and transmitting boundary condition.
  23. 04/11/2018          Lecture: notes        Riemann solutions for 2D / 3D problems (part 1).
  24. 04/13/2018          Lecture: notes       Riemann solutions for 2D / 3D problems  (part 2). DG code design: overview (part 1).
  25. 04/16/2018          Lecture: notes       DG code design: overview of PhyIntegrationCell (part 2).
  26. 04/18/2018          Lecture: notes       DG code design: overview of PhyIntegrationCell (part 3).
  27. 04/23/2018          Lecture: notes       DG code design: Different tensor storage classes at a quadrature point.
  28. 04/25/2018          Lecture: notes       DG code design: Different tensor storage classes at a quadrature point (part 2) / Integration routine.

Useful Documents

  1. Comparison of DG and CFEMs (link)

References:

  1. LeVeque, Randall J. Finite volume methods for hyperbolic problems. Vol. 31. Cambridge university press, 2002.
  2. Hesthaven, Jan S., and Tim Warburton. Nodal discontinuous Galerkin methods: algorithms, analysis, and applications. Springer Science & Business Media, 2007.
  3. Li, Ben Q. Discontinuous finite elements in fluid dynamics and heat transfer. Springer Science & Business Media, 2006.
  4. Riviere, Beatrice. Discontinuous Galerkin methods for solving elliptic and parabolic equations: theory and implementation. Society for Industrial and Applied Mathematics, 2008.