Numerical Analysis of Higher Order Discontinuous Galerkin Finite Element Methods

Kurzfassung

After the introduction in Section 1 this lecture starts off with recalling 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 in Section 2 and for the standard and the streamline diffusion finite element method of the linear advection equation in Section 3. <br> </br> We then introduce the discontinuous Galerkin discretization of the linear advection equation in Section 4. Following [Brezzi-Marini-Süli-2004] we consider two numerical flux functions, the mean-value flux and the upwind flux, and derive the corresponding <em>a priori</em> error estimates. Whereas the standard Galerkin discretization of the linear advection equation is unstable and requires e.g. streamline diffusion for stabilization, we will see in Section 4 that the discontinuous Galerkin discretization of the linear advection based on upwind is stable without addition of streamline diffusion. <br> </br> Then in Section 5, 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> 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 Section 6. 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 in Sections 6.3 and 6.4 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. <br> </br> Then in Section 7 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> This lecture is finalized with the Sections 8 and 9 which introduce the DG discretizations of the compressible Euler and Navier-Stokes equations. Additionally, the consistency and adjoint consistency analysis which has been introduced in Section 6 for linear problems is now generalized to nonlinear problems in Section 8.5. This analysis is performed for the compressible Euler and Navier-Stokes equations in Sections 8.6 and 9.3, respectively. 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> Various examples in Sections 5.6, 7.3, 8.7 and 9.4 illustrate the numerical methods described. <br> </br> In particular, the contents of this lecture is given as follows <br> </br> <br> </br> 1) Introduction <br> </br> 1.1) Higher order discretization methods <br> </br> 1.2) Discontinuous Galerkin discretizations <br> </br> 1.3) Numerical analysis of finite element methods <br> </br> 1.4) Outline <br> </br> <br> </br> 2) Higher order continuous FE methods for Poisson's equation <br> </br> 2.1) Poisson's equation <br> </br> 2.1.1) The homogeneous Dirichlet problem <br> </br> 2.1.2) The inhomogeneous Dirichlet problem <br> </br> 2.1.3) The Neumann problem <br> </br> 2.2) The standard finite element method for Poisson's equation <br> </br> 2.2.1) Consistency <br> </br> 2.2.2) Existence and uniqueness of discrete solutions <br> </br> 2.2.3) Best approximation property <br> </br> 2.2.4) Interpolation estimates <br> </br> 2.2.5) <em>A priori</em> error estimates in the H¹- and L²-norm <br> </br> <br> </br> 3) Higher order continuous FE methods for the linear advection equation <br> </br> 3.1) The linear advection equation <br> </br> 3.1.1) Variational formulation with strong boundary conditions <br> </br> 3.1.2) Variational formulation with weak boundary conditions <br> </br> 3.2) The standard Galerkin method with weak boundary conditions <br> </br> 3.3) The streamline diffusion method with weak boundary conditions <br> </br> <br> </br> 4) Higher order DG discretizations of the linear advection equation <br> </br> 4.1) Mesh related function spaces <br> </br> 4.2) A variational formulation of the linear advection equation <br> </br> 4.3) Consistency, conservation property, coercivity and stability <br> </br> 4.4) The discontinuous Galerkin discretization <br> </br> 4.5) The local L²-projection and approximation estimates <br> </br> 4.6) <em>A priori</em> error estimates <br> </br> 4.7) The discontinuous Galerkin discretization based on upwind <br> </br> 4.7.1) The importance of the inter-element jump terms <br> </br> 4.7.2) The global and local conservation property <br> </br> 4.7.3) Consistency <br> </br> <br> </br> 5) Higher order DG discretizations of Poisson's equation <br> </br> 5.1) The system and primal flux formulation <br> </br> 5.2) The DG discretization: Consistency and adjoint consistency <br> </br> 5.3) Derivation of various DG discretization methods <br> </br> 5.3.1) The SIPG and NIPG methods and the method of Baumann-Oden <br> </br> 5.3.2) The original DG discretization of Bassi and Rebay (BR1) <br> </br> 5.3.3) The modified DG discretization of Bassi and Rebay (BR2) <br> </br> 5.4) Consistency, adjoint consistency, continuity and coercivity <br> </br> 5.5) <em>A priori</em> error estimates <br> </br> 5.6) Numerical results <br> </br> <br> </br> 6) Consistency and adjoint consistency for linear problems <br> </br> 6.1) Definition of consistency and adjoint consistency <br> </br> 6.2) The consistency and adjoint consistency analysis <br> </br> 6.3) Adjoint consistency analysis of the IP discretization <br> </br> 6.3.1) The continuous adjoint problem to Poisson's equation <br> </br> 6.3.2) Primal residual form of the interior penalty DG discretization <br> </br> 6.3.3) Adjoint residual form of the interior penalty DG discretization <br> </br> 6.4) Adjoint consistency analysis of the upwind DG discretization <br> </br> 6.4.1) The continuous adjoint problem to the linear advection equation <br> </br> 6.4.2) Primal residual form of the DG discretization based on upwind <br> </br> 6.4.3) Adjoint residual form of the DG discretization based on upwind <br> </br> <br> </br> 7) <em>A priori</em> error estimates for target functionals J(.) <br> </br> 7.1) Upwind DG of the linear advection equation: Estimates in J(.) <br> </br> 7.2) IP DG discretization for Poisson's equation: Estimates in J(.) <br> </br> 7.3) Numerical results <br> </br> <br> </br> 8) Discontinuous Galerkin discretizations of the compressible Euler equations <br> </br> 8.1) Hyperbolic conservation equations <br> </br> 8.2) The compressible Euler equations <br> </br> 8.3) The DG discretization of the compressible Euler equations <br> </br> 8.4) Boundary conditions <br> </br> 8.5) Consistency and adjoint consistency for nonlinear problems <br> </br> 8.5.1) The consistency and adjoint consistency analysis <br> </br> 8.6) Adjoint consistency analysis of DG for the compressible Euler equations <br> </br> 8.6.1) The continuous adjoint problem to the compressible Euler equations <br> </br> 8.6.2) Primal residual form of DG for the compressible Euler equations <br> </br> 8.6.3) Adjoint residual form of DG for the compressible Euler equations <br> </br> 8.7) Numerical results <br> </br> <br> </br> 9) DG discretizations of the compressible Navier-Stokes equations <br> </br> 9.1) The compressible Navier-Stokes equations <br> </br> 9.2) DG discretizations of the compressible Navier-Stokes equations <br> </br> 9.3) Adjoint consistency analysis of DG for the compressible Navier-Stokes equations <br> </br> 9.3.1) The continuous adjoint problem to the compressible NS equations <br> </br> 9.3.2) Primal residual form of DG for the compressible NS equations <br> </br> 9.3.3) Adjoint residual form of DG for the compressible NS equations <br> </br> 9.4) Numerical results <br> </br> <br> </br> Acknowledgements <br> </br> Bibliography