References

Signed-off-by: Umberto Zerbinati <[email protected]>

paper/paper.bib

@@ -135,3 +135,22 @@ @TechReport{ml
   ADDRESS = "Albuquerque, NM (USA)",
   YEAR = "2004"
 }
+@article{Boffi,
+author = {D. Boffi},
+title = {Stability of higher order triangular hood-taylor methods for the stationary stokes equations},
+journal = {Mathematical Models and Methods in Applied Sciences},
+volume = {04},
+number = {02},
+pages = {223-235},
+year = {1994},
+}
+@article{HT,
+title = {A numerical solution of the Navier-Stokes equations using the finite element technique},
+journal = {Computers & Fluids},
+volume = {1},
+number = {1},
+pages = {73-100},
+year = {1973},
+issn = {0045-7930},
+author = {C. Taylor and P. Hood},
+}

paper/paper.md

@@ -71,7 +71,7 @@ $$
 \nu\Delta \vec{u} +\vec{b}\cdot \nabla\vec{u} - \nabla p = \vec{f},
 \\ \quad \nabla \cdot \vec{u} = 0,
 $$
-We discretise this problem using high-order Hood-Taylor elements ($P_4$-$P_3$) on a unit square domain. We employ an augmented Lagrangian formulation to better enforce the incompressibility constraint. We present the performance of a two level additive Schwarz preconditioner with vertex-patch smoothing as fine level correction [@BenziOlshanskii]. This preconditioner was built using ngsPETSc. The result for different viscosities $\nu$ are shown in Table 2, exhibiting reasonable robustness as the viscosity (and hence Reynolds number) changes. The full example, with more details, can be found in the [ngsPETSc documentation](https://ngspetsc.readthedocs.io/en/latest/PETScPC/oseen.py.html).
+We discretise this problem using high-order Hood-Taylor elements ($P_4$-$P_3$) on a unit square domain [@HT; @Boffi]. We employ an augmented Lagrangian formulation to better enforce the incompressibility constraint. We present the performance of a two level additive Schwarz preconditioner with vertex-patch smoothing as fine level correction [@BenziOlshanskii; @FarrellEtAll]. This preconditioner was built using ngsPETSc. The result for different viscosities $\nu$ are shown in Table 2, exhibiting reasonable robustness as the viscosity (and hence Reynolds number) changes. The full example, with more details, can be found in the [ngsPETSc documentation](https://ngspetsc.readthedocs.io/en/latest/PETScPC/oseen.py.html).
 
 Ref. Levels (N. DoFs) | $\nu=10^{-2}$|$\nu=10^{-3}$|$\nu=10^{-4}$|
 ----------------------|--------------|-------------|-------------|
@@ -90,7 +90,7 @@ Figure 2 shows a high-order NETGEN mesh employed in Firedrake for the simulation
 ![A hyperelastic beam deformed by fixing one end and applying a twist at the other end. The colouring corresponds to the deviatoric von Mises stress experienced by the beam. The beam is discretised with $P_3$ finite elements and the nonlinear problem is solved using PETSc SNES. The full example, with more details, can be found in the [ngsPETSc documentation](https://ngspetsc.readthedocs.io/en/latest/PETScSNES/hyperelasticity.py.html).](figures/hyperelastic.png)
 
 
-![Flow past a cylinder. The Navier-Stokes equations are discretised on a NETGEN high-order mesh with Firedrake. We use high-order Taylor-Hood elements ($P_4$-$P_3$) and a vertex-patch smoother as fine level correction in a two-level additive Schwarz preconditioner, [@BenziOlshanskii]. The full example, with more details, can be found in [ngsPETSc documentation](https://github.com/NGSolve/ngsPETSc). On the right a zoom near the cylinder shows the curvature of the mesh.](figures/flow_past_a_cylinder.png)
+![Flow past a cylinder. The Navier-Stokes equations are discretised on a NETGEN high-order mesh with Firedrake. We use high-order Taylor-Hood elements ($P_4$-$P_3$) and a vertex-patch smoother as fine level correction in a two-level additive Schwarz preconditioner, [@BenziOlshanskii; @FarrellEtAll]. The full example, with more details, can be found in [ngsPETSc documentation](https://github.com/NGSolve/ngsPETSc). On the right a zoom near the cylinder shows the curvature of the mesh.](figures/flow_past_a_cylinder.png)
 
 
 ![An adaptive scheme applied to the Poisson problem on an L-shaped domain. The domain is discretised using $P_1$ finite elements and the adaptive mesh refinement is driven by a Babuška-Rheinboldt error estimator [@BabuskaRheinboldt]. The adaptive procedure delivers optimal scaling of the energy norm of the error in terms of the number of degrees of freedom. The full example, with more details, can be found in the [ngsPETSc documentation](https://ngspetsc.readthedocs.io/en/latest/utils/firedrake/lomesh.py.html).](figures/adaptivity.png)