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paper/paper.md

@@ -54,7 +54,7 @@ ngsPETsc aims to assist with the solution of challenging PDEs on complex geometr
 # Examples
 
 In this section we provide a few examples of results that can be obtained using ngsPETSc.
-We begin considering a simple Poisson problem on a unit square domain discretised with $P_2$ finite elements and compare the performance of different solvers available in NGSolve via ngsPETSc. In particular we compared PETSc GAMG [@PETScGAMG], PETSc BDDC [@PETScBDDC] and NGSolve own implementation of element-wise BDDC. The result is shown in Table 1 and the full example, with more details, can be found in the [ngsPETSc documentation](https://ngspetsc.readthedocs.io/en/latest/PETScKSP/poisson.py.html).
+We begin by considering a simple Poisson problem on a unit square domain discretised with $P_2$ finite elements and compare the performance of different solvers available in NGSolve via ngsPETSc. In particular we compared PETSc's algebraic multigrid algorithm GAMG [@PETScGAMG], PETSc's domain decomposition BDDC algorithm [@PETScBDDC] and NGSolve's own implementation of element-wise BDDC. The result is shown in Table 1 and the full example, with more details, can be found in the [ngsPETSc documentation](https://ngspetsc.readthedocs.io/en/latest/PETScKSP/poisson.py.html).
 
 N. DoFs  | PETSc GAMG   | PETSc BDDC (N=2) | PETSc BDDC (N=4) | PETSc BDDC (N=6) | Element-wise BDDC* |
 ---------|--------------|------------------|------------------|------------------|--------------------|
@@ -64,12 +64,12 @@ N. DoFs  | PETSc GAMG   | PETSc BDDC (N=2) | PETSc BDDC (N=4) | PETSc BDDC (N=6)
 
 Table 1: The number of degrees of freedom (DoFs) and the number of iterations required to solve the Poisson problem with different solvers. The numbers in parentheses are the relative residuals. *Element-wise BDDC is a custom implementation of BDDC preconditioner in NGSolve.
 
-We then consider the Oseen problem, i.e.
+We next consider the Oseen problem, i.e.
 $$
 \nu\Delta \vec{u} +\vec{b}\cdot \nabla\vec{u} - \nabla p = \vec{f},
 \\ \quad \nabla \cdot \vec{u} = 0,
 $$
-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 and 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. 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).
 
 Ref. Levels (N. DoFs) | $\nu=10^{-2}$|$\nu=10^{-3}$|$\nu=10^{-4}$|
 ----------------------|--------------|-------------|-------------|
@@ -83,13 +83,13 @@ Figure 1 shows a simulation of a hyperelastic beam, solved with PETSc nonlinear
 Figure 2 shows a high-order NETGEN mesh employed in Firedrake for the simulation of a Navier-Stokes flow past a cylinder. Figure 3 shows the adaptive mesh refinement for a Poisson problem on an L-shaped domain.
 
 
-![A hyperelastic beam deformed by fixing one end and applying a twist at the other end. The coloruing 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)
+![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 and 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]. 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 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)
+![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)
 
 
-More example can be found in the documentation of ngsPETSc manual [@manual].
+More examples can be found in the documentation of ngsPETSc manual [@manual].