Numerical Analysis of Higher Order Discontinuous Galerkin Finite Element methods

Kurzfassung

In part 1 of this lecture we recall well-known results from the numerical analysis of the continuous finite element methods. In particular, we recall <em>a priori</em> error estimates in the energy norm and the L²-norm including their proofs for higher order standard finite element methods of Poisson's equation and for the standard and the streamline diffusion finite element method of the linear advection equation. Then we follow [Arnold-Brezzi-Cockburn-Marini-2002] and derive and analyze a variety of discontinuous Galerkin discretizations of Poisson's equations. In particular, we derive the symmetric and non-symmetric interior penalty Galerkin method (SIPG and NIPG), the method of Baumann-Oden (BO) and the first and second method of Bassi and Rebay (BR1 and BR2). The analysis of the methods includes the consistency and adjoint consistency of the schemes, continuity and coercivity of the respective bilinear forms and <em>a priori</em> error estimates for the interior penalty methods. In particular, we will see that the adjoint consistent SIPG scheme is of optimal order in the L²-norm whereas the adjoint inconsistent NIPG scheme is not. <br> </br> <br> </br> Motivated by the connection of adjoint consistency of DG discretizations to the availability of optimal order error estimates in the L²-norm we concentrate on the adjoint consistency property in lecture part 2. In particular, here we follow [Hartmann-2007] and give a general framework for analyzing the consistency and adjoint consistency of DG discretizations for linear problems with inhomogeneous boundary conditions. This includes the derivation of continuous adjoint problems associated to specific target quantities, the derivation of primal and adjoint residual forms of the discretizations and the discussion whether the discretizations in combination with specific target quantities J(.) are adjoint consistent or not. This analysis is performed for the interior penalty DG discretization of the Dirichlet-Neumann boundary value problem of Poisson's equations and for the upwind DG discretization of the linear advection equation, respectively. Then, the previously shown properties and estimates for the interior penalty and the upwind DG discretization are used to derive <em>a priori</em> estimates for the error measured in terms of target quantities J(.). Here again, we will see that a discretization must be consistent and adjoint consistent in order to provide optimal error estimates in J(.). <br> </br> <br> </br> This lecture is finalized in lecture part 3 which introduces the DG discretizations of the compressible Euler and Navier-Stokes equations. Additionally, the consistency and adjoint consistency analysis which has been introduced in lecture part 2 for linear problems is now generalized to nonlinear problems. This analysis is performed for the compressible Euler and Navier-Stokes equations. This includes the derivation of an adjoint consistent discretization of boundary conditions and of target functionals. Here particular emphasis is placed on the aerodynamic force coefficients like the drag, lift and moment coefficients. <br> </br> <br> </br> Various examples in lecture parts 1,2, and 3 illustrate the numerical methods described.