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Special Radon Semester on Computational Mechanics
Linz, October-December 2005
"Discontinuous Galerkin methods" organized by Raytcho Lazarov

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Main Topics with Abstracts

  1. The origin of DG methods (RL)(90 min):
    Abstract. Here we shall cover the origin of the DG methods that will include: (1) 1-D transport equation; (2) multidimensional transport equations and application to the neutron transport system (combined with source iteration) (3) time discretization of parabolic problems.
    References: [10], [24], [33]

  2. DG as stabilization of FEM for problem with Lagrange multipliers (RL) (90 min):
    Abstract. Here we shall consider elliptic problems when imposing the Dirichlet boundary conditions as constrains, Stokes equations and incompressible elasticity via DG method
    References: [2], [19], [35], [29], [32]

  3. DG methods for convection-diffusion problems and conservation laws (RL) (90 min):
    Abstract. Here we shall present various strategies for approximation of convection-diffusion equations with emphasis on the boundary conditions, stable flux approximation (up-wind or Lax-Friedrichs fluxes) and applications
    References: [4], [10], [17]

  4. DG methods for mixed problems (RL) (90 min):
    Abstract. We present mixed formulation of second order elliptic problems, Stokes equations, and the equations of incompressible elasticity; (for elliptic equations partially this should have been discussed in Lecture 8); for Stokes and incompressible elasticity we shall follow the paper [23] and add the papers [14,15] which introduce a very general framework for DG methods;
    References: [1], [23], [14], and [15]

  5. DG methods in solving Euler/Navier-Stokes equations (RL) (90 min):
    Abstract. This is a very large topic, we should concentrate to some basic model problems, from Stokes, Oseen, and Navier-Stokes equations;
    References: [3], [5], [13], [21], [27], [30], [31].

  6. DG methods in solving Euler/Navier-Stokes equations (RL) (90 min):
    Abstract. Continuation of the previouys lecture 5.

  7. General stabilization mechanisms of DG FEM (ST) (90 min):
    Abstract. We discuss in more abstract form the stabilization mechanisms of DG FEM This will follow the paper [9] by F. Brezzi, B. Cockburn, D.L. Marini, and E. Süli;
    References: [9]

  8. Unified analysis of DG methods for elliptic problem: I (ST) (90 min):
    Abstract. Here we shall present an attempt to unify the DG approximations for second order elliptic problems by introducing the concept of numerical fluxes ( follow first three sections of [1])
    References: [1] (for $ p$-version, [34])

  9. Unified analysis of DG methods for elliptic problem: II (ST) (90 min):
    Abstract. Here we shall present analysis of the DG approximations for second order elliptic problems: boundedness, stability, consistency, error estimates in $ H^1$ and $ L^2$ norms (Sections 4 and 5 of [1])
    References: [1] (for $ p$-version, [34])

  10. MG for symmetric DG systems (RL/UL) (90 min):
    Abstract. We present MG method as a preconditioner for DG systems for second order elliptic problems; this is the classical MG method (V-, W-, and F-cycles) for geometrically nested meshes studied in [22] and [8]. This might be a research topic as well.
    References: [22], [8].

  11. Preconditioning of DG systems (RL/JK) (90 min):
    Abstract. Preconditioning for DG systems for second order elliptic problems and Stokes equations that include the MG method as well as some recent considerations having more algebraic flavour of [28]. It would be nice to look at fourth order problems as well. This might be a research topic as well.
    References: [22], [8], [6], and [28].

  12. Hybridization of the DG FEM (RL) (90 min):
    Abstract. This technique is applied to mixed DG methods for second order problems. As in the case of standard mixed FEM the goal here is via suitable Lagrange multipliers to locally eliminate the original unknowns and reduce the problem to an approximation of the elliptic problem by the multipliers, I have in mind to follow the paper [11].
    References: [11], [12]

  13. DG methods for fourth order problems (RL) (90 min):
    Abstract. We present the results from [6] on fourth order elliptic problems; (this is a stand alone subject that could be scheduled after most of the material is covered)
    References: [6]


Next: Tentative Schedule for Lectures Up: L_Schedule Previous: L_Schedule
Satyendra Tomar 2005-08-18

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