Q2. What are the future works in "An optimal order interior penalty discontinuous galerkin discretization of the compressible navier–stokes equations" ?
Future work will be devoted to the application of the scheme to both three–dimensional laminar and turbulent flows.
Q3. What is the definition of the lift and drag coefficients?
The lift and drag coefficients are defined as integrals, over the boundary of the body, of the viscous and pressure induced forces normal and tangential to the flow, respectively.
Q4. What is the main advantage of DGFEMs?
there is considerable flexibility in the choice of the mesh design; indeed, DGFEMs can easily handle non-matching grids and non-uniform, even anisotropic, polynomial approximation degrees.
Q5. How many edges must be chosen for a given element?
CBR2 must be chosen to be at least as large as the total number of edges a given element possesses, i.e., in the case of quadrilateral meshes, CBR2 ≥ 4.
Q6. How much computational effort is required for the evaluation of the nonlinear residual?
The computation of the lifting operator present within many DGFEMs is typically quite expensive; indeed, for a nonlinear problem, around 30% of the computational effort required for the evaluation of the nonlinear residual is devoted to the computation of the lifting operator.
Q7. How is the interior penalty DGFEM used?
the use of, for example, the nonsymmetric interior penalty method for the numerical approximation of the Poisson’s equation leads to suboptimal orders of convergence when the error is measured in terms of both the L2-norm, as well as for target functionals of the solution, cf. [23].
Q8. What is the purpose of this paper?
Building on the techniques outlined in their previous articles [25,27,28], the aim of this paper is to propose a new alternative interior penalty DGFEM for the numerical approximation of the compressible Navier–Stokes equations which leads to computationally optimal orders of convergence when the error is computed in terms of both the L2-norm, as well as for certain target functionals of the solution of practical interest.
Q9. What is the reason for the sub-optimal convergence observed when using these two schemes?
The sub-optimal convergence observed when employing these two schemes is attributed to the lack of smoothness in the resulting dual problems, cf. [23].
Q10. What is the effectivity index of the mesh generated for the calculation of the drag coefficient?
As noted in [27], refinement is mainly concentrated within in the vicinity of the airfoil, with the mesh generated for thecomputation of the lift coefficient being more concentrated around the the airfoil, than the corresponding mesh generated for the accurate computation of the drag coefficient.
Q11. What is the tensor product polynomial of degree p?
On the reference element κ̂ the authors define the space of tensor product polynomials of degree p ≥ 0 as follows:Qp(κ̂) = span {x̂α : 0 ≤ αi ≤ p, 1 ≤ i ≤ 2} ,where α denotes a multi-index and x̂α = ∏2i=1 x̂ αi i .
Q12. What is the way to calculate the interior penalty DGFEM?
as the authors shall see later in this article, in the context of duality–based error estimation, the symmetric version of the interior penalty DGFEM proposed in [27] may not lead to optimal rates of convergence as the mesh size tends to zero.
Q13. How is the new scheme able to yield optimal rates of convergence?
the newly proposed scheme has been shown to yield optimal rates of convergence, when the error is measured in terms of both the L2-norm, as well as for certain target functionals of thesolution of practical interest.
Q14. What is the tensor of the f v(u,u)?
Writing G to denote the homogeneity tensor, with Gij(u) = ∂f v i (u,∇u)/∂uxj , for i, j = 1, 2, cf. [27], the viscous fluxes may be written in the form f vi (u,∇u) = Gij(u)∂u/∂xj , i = 1, 2, or more compactly, the authors may write F v(u,∇u) = G(u)∇u.
Q15. What is the common type of lifting operator?
it is worth noting that for one prominent class of DGFEMs, referred to as the interior penalty DGFEMs, the lifting operator may be explicitly evaluated; indeed, here the lifting operator simply reduces to the identity operator.