An optimal order interior penalty discontinuous Galerkin discretization of the compressible Navier-Stokes equations

In this article we propose a new symmetric version of the interior penalty discontinuous Galerkin finite element method for the numerical approximation of the compressible Navier-Stokes equations. Here, particular emphasis is devoted to the construction of an optimal numerical method for the evaluation of certain target functionals of practical interest, such as the lift and drag coefficients of a body immersed in a viscous fluid. With this in mind, the key ingredients in the construction of the method include: (i) An adjoint consistent imposition of the boundary conditions; (ii) An adjoint consistent reformulation of the underlying target functional of practical interest; (iii) Design of appropriate interior--penalty stabilization terms. Numerical experiments presented within this article clearly indicate the optimality of the proposed method when the error is measured in terms of both the L2-norm, as well as for certain target functionals. Computational comparisons with other discontinuous Galerkin schemes proposed in the literature, including the second scheme of Bassi and Rebay, the standard SIPG method outlined in [Hartmann,Houston-2006], and an NIPG variant of the new scheme will be undertaken

An A Posteriori Error Indicator for Discontinuous Galerkin Approximations of Fourth Order Elliptic Problems

We introduce a residual-based a posteriori error indicator for discontinuous Galerkin discretizations of the biharmonic equation with essential boundary conditions. We show that the indicator is both reliable and efficient with respect to the approximation error measured in terms of a natural energy norm, under minimal regularity assumptions. We validate the performance of the indicator within an adaptive mesh refinement procedure and show its asymptotic exactness for a range of test problems

Symmetric Interior Penalty DG Methods for the Compressible Navier-Stokes Equations II: Goal--Oriented A Posteriori Error Estimation

In this article we consider the application of the generalization of the symmetric version of the interior penalty discontinuous Galerkin finite element method to the numerical approximation of the compressible Navier--Stokes equations. In particular, we consider the a posteriori error analysis and adaptive mesh design for the underlying discretization method. Indeed, by employing a duality argument (weighted) Type I a posteriori bounds are derived for the estimation of the error measured in terms of general target functionals of the solution; these error estimates involve the product of the finite element residuals with local weighting terms involving the solution of a certain dual problem that must be numerically approximated. This general approach leads to the design of economical finite element meshes specifically tailored to the computation of the target functional of interest, as well as providing efficient error estimation. Numerical experiments demonstrating the performance of the proposed approach will be presented

Two-grid hp-version discontinuous Galerkin finite element methods for quasi-Newtonian fluid flows

In this article we consider the a priori and a posteriori error analysis of two-grid hp-version discontinuous Galerkin finite element methods for the numerical solution of a strongly monotone quasi-Newtonian fluid flow problem. The basis of the two-grid method is to first solve the underlying nonlinear problem on a coarse finite element space; a fine grid solution is then computed based on undertaking a suitable linearization of the discrete problem. Here, we study two alternative linearization techniques: the first approach involves evaluating the nonlinear viscosity coefficient using the coarse grid solution, while the second method utilizes an incomplete Newton iteration technique. Energy norm error bounds are deduced for both approaches. Moreover, we design an hp-adaptive refinement strategy in order to automatically design the underlying coarse and fine finite element spaces. Numerical experiments are presented which demonstrate the practical performance of both two-grid discontinuous Galerkin methods

Adaptive discontinuous Galerkin methods on polytopic meshes

In this article we consider the application of discontinuous Galerkin finite element methods, defined on agglomerated meshes consisting of general polytopic elements, to the numerical approximation of partial differential equation problems posed on complicated geometries. Here, we assume that the underlying computational domain may be accurately represented by a geometry-conforming fine mesh; the resulting coarse mesh is then constructed based on employing standard graph partitioning algorithms. To improve the accuracy of the computed numerical approximation, we consider the development of goal-oriented adaptation techniques within an automatic mesh refinement strategy. In this setting, elements marked for refinement are subdivided by locally constructing finer agglomerates; should further resolution of the underlying fine mesh T_f be required, then adaptive refinement of T_f will also be undertaken. As an example of the application of these techniques, we consider the numerical approximation of the linear elasticity equations for a homogeneous isotropic material. In particular, the performance of the proposed adaptive refinement algorithm is studied for the computation of the (scaled) effective Young's modulus of a section of trabecular bone

Symmetric Interior Penalty DG Methods for the Compressible Navier-Stokes Equations I: Method Formulation

In this article we consider the development of discontinuous Galerkin finite element methods for the numerical approximation of the compressible Navier-Stokes equations. For the discretization of the leading order terms, we propose employing the generalization of the symmetric version of the interior penalty method, originally developed for the numerical approximation of linear self-adjoint second-order elliptic partial differential equations. In order to solve the resulting system of nonlinear equations, we exploit a (damped) Newton-GMRES algorithm. Numerical experiments demonstrating the practical performance of the proposed discontinuous Galerkin method with higher-order polynomials are presented

Two-grid hp-DGFEM for second order quasilinear elliptic PDEs based on an incomplete Newton iteration

In this paper we propose a class of so-called two-grid hp-version discontinuous Galerkin finite element methods for the numerical solution of a second-order quasilinear elliptic boundary value problem based on the application of a single step of a nonlinear Newton solver. We present both the a priori and a posteriori error analysis of this two-grid hp-version DGFEM as well as performing numerical experiments to validate the bounds

Discontinuous Galerkin finite element approximation of quasilinear elliptic boundary value problems II: strongly monotone quasi-Newtonian flows

In this article we develop both the a priori and a posteriori error analysis of hp– version interior penalty discontinuous Galerkin finite element methods for strongly monotone quasi-Newtonian fluid flows in a bounded Lipschitz domain Ω ⊂ R^d, d = 2, 3. In the latter case, computable upper and lower bounds on the error are derived in terms of a natural energy norm which are explicit in the local mesh size and local polynomial degree of the approximating finite element method. A series of numerical experiments illustrate the performance of the proposed a posteriori error indicators within an automatic hp–adaptive refinement algorithm