# The 2D annulus benchmark

*This section was contributed by C. Thieulot, E. G. Puckett, and R. Gassmoeller.*

This benchmark is based on a manufactured solution in which an analytical
solution to the isoviscous incompressible Stokes equations is derived in an
annulus geometry. The velocity and pressure fields are as follows:
```{math}
v_r(r,\theta)     &=  g(r) k \sin(k\theta), \\
v_\theta(r,\theta)&=  f(r) \cos(k \theta), \\
p(r,\theta)       &=  k h(r) \sin(k \theta) + \rho_0 (R2-r), \\
\rho (r,\theta)   &=  m(r) k \sin (k \theta) + \rho_0,
```
with
```{math}
f(r)&=Ar+B/r, \\
g(r) &= \frac{A}{2}r  +  \frac{B}{r} \ln r + \frac{C}{r}, \\
h(r)&= \frac{2g(r)-f(r)}{r},  \\
m(r) &= g'' - \frac{g'}{r}  - \frac{g}{r^2} (k^2 - 1)  + \frac{f}{r^2}   + \frac{f'}{r}, \\
A &= -C\frac{2(\ln R_1 - \ln R_2)} { R_2^2 \ln R_1  - R_1^2 \ln R_2}, \\
B &= -C \frac{R_2^2-R_1^2}{R_2^2 \ln R_1 - R_1^2 \ln R_2}.
```
The parameters $A$ and $B$ are chosen so that $v_r(R_1)=v_r(R_2)=0$, i.e. the
velocity is tangential to both inner and outer surfaces. The gravity vector is
radial inward and of unit length.

The parameter $k$ controls the number of convection cells present in the
domain, as shown in {numref}`fig:annulus-vp1` through {numref}`fig:annulus-vp3`.


```{figure-md} fig:annulus-vp1
<img src="pressures.png" width="100%"/>

 Pressure fields for k=0,1,2,4 for the 2D annulus benchmark.
```

```{figure-md} fig:annulus-vp2
<img src="density.png" width="100%"/>

Density fields for k=0,1,2,4 for the 2D annulus benchmark.
```

```{figure-md} fig:annulus-vp3
<img src="velocities2.png" width="100%"/>

Velocity fields for k=0,1,2,4 for the 2D annulus benchmark.
```

In the present case, we set $R_1=1$, $R_2=2$ and $C=-1$. {numref}`fig:annulus-errors` shows
the velocity and pressure errors in the $L_2$-norm as a function of the mesh
size $h$ (taken in this case as the radial extent of the elements). As
expected we recover a third-order convergence rate for the velocity and a
second-order convergence rate for the pressure.

```{figure-md} fig:annulus-errors
<img src="errors_annulus.svg" />

Velocity and pressure errors in the <span class="math inline"><em>L</em><sub>2</sub></span>-norm as a function of the mesh size for the 2D annulus benchmark.
```

Additionally, the subdirectory
[benchmarks/annulus/transient](https://github.com/geodynamics/aspect/tree/main/benchmarks/annulus/transient)
contains an extension of the benchmark
to time-dependent flow. The benchmark and its results are described in
Gassmoeller et al. (2023), "Benchmarking the accuracy of higher order
particle methods in geodynamic models of transient flow", see there
for a detailed description.