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 2/19/2020).
  • 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 2/26/2020
      • Part 2: due 3/2/2020
    • HW4: (link) Parabolic PDEs
      • Part 1, due 3/5/2020
      • Part 2: due 3/12/2020
    • HW5: (link) Hyperbolic PDEs
      • Part 1, due 3/19/2020
      • Part 2: due 3/26/2020
    • HW6: (link) SDG method, Due 4/9/2020
    • HW7: (link) Riemann solutions, Due 4/18/2020

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

  1. notes, video Comparison of continuous and discontinuous Galerkin FEMs. Sample thermal problem: CFEM formulation (part 1).
  2. notes, video Comparison of continuous and discontinuous Galerkin FEMs. Sample thermal problem: CFEM formulation (part 2).
  3. notes, video Sample thermal problem: CFEM formulation / DG formulation (part 1).
  4. notes, video Sample thermal problem: DG formulation (part 2). Explicit versus Implicit time integration for a parabolic PDE / DG vs. CFEM.
  5. notes, video DG vs. CFEM.
  6. Notes, video DG fluxes and relation to interior penalty methods (part 1).
  7. Notes, video-1 video-2 DG fluxes and relation to interior penalty methods (part 2).
  8. Notes, video   DG fluxes and relation to interior penalty methods (part 3).
  9. Notes, video   Notes on DG method for elliptic and parabolic PDEs (part 1).
  10. Notes, video   Notes on DG method for elliptic and parabolic PDEs (part 2).
  11. Notes, video   DG formulation for hyperbolic PDEs (part 1).
  12. Notes, video  Parabolic PDEs: physical flux formulas, erroneous flux option; DG formulation for hyperbolic PDEs (part 2).
  13. Notes, video DG formulation for hyperbolic PDEs (part 3).
  14. Notes, video DG formulation for hyperbolic PDEs (part 4).
  15. Notes, video Comparison of 1- and 2-field formulations for hyperbolic PDEs; relation to Helmholtz equation; Spacetime methods: expression of balance laws in spacetime (part 1).
  16. Notes, video Spacetime methods: expression of balance laws in spacetime (part 2); different spacetime methods.
  17. Notes, video Spacetime methods: Comparison of different methods; Weighted residual and weak forms for a hyperbolic heat equation (part 1).
  18. Notes, video Spacetime methods: Comparison of different methods; Weighted residual and weak forms for a hyperbolic heat equation (part 2). Global stability using the spectral radius.
  19. Notes, video Spacetime Discontinous Galerkin methods (part 2): Global stability analysis for linear problems; numerical example of 1D problem (1 patch solved); geometric and algebraic stiffness.
    Sessions 19 and 20 use these notes from 2020S for a hand-calculation of a patch (link) and discussion of some properties of SDGs (link).
  20. Notes, video Spacetime Discontinuous Galerkin methods (part 3): Comparison with other methods, adaptivity in spacetime. Transfer matrix and spectral stability analysis.
    Sessions 19 and 20 use these notes from 2020S for a hand-calculation of a patch (link) and discussion of some properties of SDGs (link).
  21. Notes, video Spacetime Discontinuous Galerkin methods (part 4): von Neumann stability and dispersion error analysis.
  23. 03/30/2020         Lecture: notes     video Spacetime Discontinous Galerkin methods (part 2).
  24. 04/01/2020         Lecture: notes     video Spacetime Discontinous Galerkin methods (part 3); properties of cSDG method.
  25. 04/08/2020         Lecture: notes     video Riemann solutions for 2D / 3D problems (part 1).
  26. 04/13/2020         Lecture: notes     video Riemann solutions for 2D / 3D problems (part 2).
  27. 04/15/2020         Lecture: notes     video Riemann and approximate Riemann solutions for nonlinear conservation laws (part 1).
  28. 04/20/2020         Lecture: notes     video Riemann and approximate Riemann solutions for nonlinear conservation laws (part 2); Transmitting BC (Silver Muller method).
  29. 04/22/2020         Lecture: notes     video Transmitting BC (PML). DG code design: overview (part 1).
  30. 04/27/2020         Lecture: notes     video DG code design: overview of PhyIntegrationCell (part 2).
  31. 04/29/2020         Lecture: notes     video DG code design: Different tensor storage classes at a quadrature point / Integration routine.

Useful Documents

  1. Comparison of DG and CFEMs (link)
  2. Some references.

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.