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\bf{Supplement to "Practical analytical solutions for benchmarking
of 2-D and 3-D geodynamic Stokes problems with variable
viscosity"}
%
%\Author[2]{I.~Yu.}{Popov} \Author[2]{I.~S.}{Lobanov}
%\Author[2]{S.~I.}{Popov} \Author[2]{A.~I.}{Popov}
%\Author[1]{T.~V.}{Gerya}
%
%\affil[1]{Institute of Geophysics, Department of Earth Sciences,
%Swiss Federal Institute of Technology Zurich (ETH), 5
%Sonnegstrasse, 8092 Zurich, Switzerland} \affil[2]{St. Petersburg
%National Research University of Information Technologies,
%Mechanics and Optics, 49 Kronverkskiy, St. Petersburg, 197101,
%Russia}


\section{Example problems and numerical convergence tests}

The scheme of algorithm testing is conventional. We have obtained
particular  solutions of the Stokes and continuity equations for
two types of viscosity variations. Let us choose a domain, e.g., a
rectangle in 2D case. Calculate values of velocity and pressure
given by our analytical solution and take this values as the
boundary conditions. Then, due to the uniqueness theorem the
solution of the boundary problem in the domain should coincide
with our analytical solution. Let us compute the solution of the
boundary problem by a numerical method. Comparison of the result
with the exact analytical solution shows the quality of the
numerical algorithm.  In the present paper, we used standard 2D
and 3D stress-conservative finite-differences on staggered
regularly spaced grid for  obtaining numerical solutions (Gerya et
al., 2010). Respective MatLab  programs for 2D and 3D cases are
provided as supplements to this paper.


\subsection{2D example}
\subsubsection{Linearly varying viscosity}

Consider a simple example of such flow in a rectangle $0\leq x\leq
x_{size},0\leq y\leq y_{size}.$ We assume that $\eta =ax+by+c. $
We will mark the exact solution obtained in Section 2 as $v_{x,a},
v_{y,a}, P_a$. It is the solution of the boundary problem in the
rectangle $\Omega$ with the following conditions at the boundary
$\partial \Omega = \{x=0, x=x_{size}, y=0, y=y_{size}\}:$
$$
v_{y}\vert_{\partial \Omega} = v_{y,a}, \quad v_{x}\vert_{\partial
\Omega} = v_{x,a}.
$$


Let us compute the velocity and pressure with using of
finite-difference scheme. The corresponding solution is marked as
$v_{x,n}, v_{y,n}, P_n$. The deviation of these values from the
exact solution ($v_{x,n}-v_{x,a}, v_{y,n}- v_{y,a}, P_n - P_a$) is
related with the error of the numerical scheme We calculate the
relative errors of three types: $L_{\infty}, L_1, L_2 $ for
different viscosity contrasts , i.e. different values of the
coefficients $a,b$. We test the program
Stokes2D-variable-viscosity1 from \cite{G}.  The results are
presented at Fig. 1 - 6. Namely, figures 1-3 correspond to low
viscosity contrast, Figures 4-6 - to high viscosity contrast.
Particularly, Fig. 1 and Fig. 4 show pressure and velocity
components distributions. Fig. 2 and Fig. 4 characterizes the
viscosity and the density distributions. Fig. 3 and Fig. 6 contain
plots of relative errors via the grid resolutions in logarithmic
scale. The viscosity contrast, i.e. the values of the coefficients
in the expression for the viscosity, is determined by given values
of the viscosity at three rectangle corners. The value of the
viscosity at the initial rectangle corner is $1$, $\eta_2, \eta_3$
are the values of the viscosity at two adjacent corners. At all
figures $"n"$ means "numerical solution", $"a"$ means "analytical
solution" (benchmark).

For the case of linearly varying viscosity we made calculations
for the following system parameters:
$$
x_{size} = y_{size} = 1, G_x = 0, G_y = 10,
$$
$$
\eta_1 =1, \beta_1 = 1, \beta_2 = 3\times 10^3,
$$
$$
c=\eta_1, a=(\eta_3 - \eta_1)/x_{size}, b=(\eta_2 -
\eta_1)/y_{size},
$$
$$
\rho = \beta_1 (ax + by +c) + \beta_2.
$$

\begin{figure}
\centerline{\includegraphics[width=3in]{fig_1_2D_low_lin.jpg}}
 \caption{ Distribution of $v_x$, $v_y$ and $P$; 2D case,
 linearly varying viscosity, low viscosity contrast
 ($\eta_2 = \eta_3 = 5$). Here and in the following indices
 at the computed velocity components and pressure indicate
 analytical (a) and numerical (n) values and difference
 between them (e).}\label{fig_1_2D_low_lin}
\end{figure}

\begin{figure}
\centerline{\includegraphics[width=3in]{fig_4_2D_low_lin.jpg}}
 \caption{ Distribution of viscosity $\eta$ and
 density $\rho$; 2D case,
 linearly varying viscosity, low viscosity contrast
 ($\eta_2 = \eta_3 = 5$). }\label{fig_4_2D_low_lin}
\end{figure}

\begin{figure}
\centerline{\includegraphics[width=3in]{fig_3_2D_low_lin.jpg}}
 \caption{ Logarithm of the relative error via
 logarithm of the grid step; 2D case,
 linearly varying viscosity, low viscosity contrast
 ($\eta_2 = \eta_3 = 5$); blue line - pressure, green - $v_x$,
 black - $v_y$; line - $L_1$-error,
 dashed line - $L_{\infty}$-error, line with dots -
 $L_2$-error. }\label{fig_3_2D_low_lin}
\end{figure}


\begin{figure}
\centerline{\includegraphics[width=3in]{fig_1_2D_high_lin.jpg}}
 \caption{ Distribution of $v_x$, $v_y$ and $P$; 2D case,
 linearly varying viscosity, high viscosity contrast
 ($\eta_2 = \eta_3 = 100$). }\label{fig_1_2D_high_lin}
\end{figure}


\begin{figure}
\centerline{\includegraphics[width=3in]{fig_4_2D_high_lin.jpg}}
 \caption{ Distribution of viscosity $\eta$ and
 density $\rho$; 2D case,
 linearly varying viscosity, high viscosity contrast
 ($\eta_2 = \eta_3 = 100$).  }\label{fig_4_2D_high_lin}
\end{figure}

\begin{figure}
\centerline{\includegraphics[width=3in]{fig_3_2D_high_lin.jpg}}
 \caption{ Logarithm of the relative error via
 logarithm of the grid step; 2D case,
 linearly varying viscosity, high viscosity contrast
 ($\eta_2 = \eta_3 = 100$); blue line - pressure, green - $v_x$,
 black - $v_y$; line - $L_1$-error,
 dashed line - $L_{\infty}$-error, line with dots -
 $L_2$-error. }\label{fig_3_high_lin}
\end{figure}

One can see that there is rather high accuracy of the numerical
approach. We observe the conventional situation -
$L_{\infty}$-error is the largest among the considered errors
norms, and $L_1$-error and $L_2 -$error are similar. The
calculations show that one has good convergence of the numerical
scheme for small viscosity contrast, but it is not so for high
viscosity contrast (compare Fig. 3 and Fig. 6).

To describe in more details the dependence of the error on the
viscosity contrast we fill tables with errors for different values
of $\eta_2, \eta_3$ (see Appendix, tables 1-9).

\subsubsection{Exponentially varying viscosity}

The case of exponentially varying viscosity is treated
analogously. To have a possibility of comparison with the case of
linearly varying viscosity we take the same system parameters
(geometrical size, gravitational terms and the dependence of the
density on the viscosity) with the same viscosity contrast (i.e.
the values of the viscosity at the rectangle corners)

$$C=\eta_1,
    a=(\log(\eta_3)-\log(\eta_1))/x_{size},
  $$
  $$
    b=(\log(\eta_2)-\log(\eta_1))/y_{size},
$$
$$
\eta= C \exp{(a x + b y)},
$$
$$
\rho = \beta_1 \eta + \beta_2
$$
$$
 x_{size} = y_{size} = 1,
$$
$$
 G_x = 10, G_y = 10,
$$
$$
\eta_1 =1, \beta_1 = 1, \beta_2 = 3\times 10^3,
$$
We deal with two cases (low and high viscosity contrasts). Figures
7-12 present the results.  Namely, figures 7-9 correspond to the
case of low viscosity contrast, figures 10-12 are related with the
case of high viscosity contrast. There are some similarities with
the previous case. Particularly, $L_{\infty}$ error norm gives us
the maximum relative error value among three considered error
norms. We have small errors (see the velocity and pressure
distributions, Fig. 7, Fig. 10). But peculiarities are more
interesting. Namely, the numerical scheme works essentially better
for the case of exponentially varying viscosity than for linearly
varying viscosity. We observe good convergence both for low and
for high viscosity contrasts (compare Fig. 9, Fig. 12 and,
correspondingly, Fig. 3, Fig. 6).



\begin{figure}
\centerline{\includegraphics[width=3in]{fig_1_2D_low_exp.jpg}}
 \caption{ Distribution of $v_x$, $v_y$ and $P$; 2D case,
 exponentially varying viscosity, low viscosity contrast
 ($\eta_2 = \eta_3 = 5$). }\label{fig_1_2D_low_exp}
\end{figure}

\begin{figure}
\centerline{\includegraphics[width=3in]{fig_4_2D_low_exp.jpg}}
 \caption{ Distribution of viscosity $\eta$ and
 density $\rho$; 2D case,
 exponentially varying viscosity, low viscosity contrast
 ($\eta_2 = \eta_3 = 5$). }\label{fig_4_2D_low_exp}
\end{figure}

\begin{figure}
\centerline{\includegraphics[width=3in]{fig_3_2D_low_exp.jpg}}
 \caption{ Logarithm of the relative error via
 logarithm of the grid step; 2D case,
 exponentially varying viscosity, low viscosity contrast
 ($\eta_2 = \eta_3 = 5$); blue line - pressure, green - $v_x$,
 black - $v_y$; line - $L_1$-error,
 dashed line - $L_{\infty}$-error, line with dots -
 $L_2$-error. }\label{fig_3_2D_low_exp}
\end{figure}



\begin{figure}
\centerline{\includegraphics[width=3in]{fig_1_2D_high_exp.jpg}}
 \caption{ Distribution of $v_x$, $v_y$ and $P$; 2D case,
 exponentially varying viscosity, high viscosity contrast
 ($\eta_2 = \eta_3 = 100$). }\label{fig_1_2D_high_exp}
\end{figure}


\begin{figure}
\centerline{\includegraphics[width=3in]{fig_4_2D_high_exp.jpg}}
 \caption{ Distribution of viscosity $\eta$ and
 density $\rho$; 2D case,
 exponentially varying viscosity, high viscosity contrast
 ($\eta_2 = \eta_3 = 100$).  }\label{fig_4_2D_high_exp}
\end{figure}

\begin{figure}
\centerline{\includegraphics[width=3in]{fig_3_2D_high_exp.jpg}}
 \caption{ Logarithm of the relative error via
 logarithm of the grid step; 2D case,
 exponentially varying viscosity, high viscosity contrast
 ($\eta_2 = \eta_3 = 100$); blue line - pressure, green - $v_x$,
 black - $v_y$; line - $L_1$-error,
 dashed line - $L_{\infty}$-error, line with dots -
 $L_2$-error. }\label{fig_3_2D_high_exp}
\end{figure}

Dependence of the errors on the viscosity contrast for
exponentially varying viscosity is presented in the tables 10-18
in Appendix.








\subsection{3D example}

One can note that benchmark solutions for 3D case are essentially
more rare than for the corresponding 2D situation. It is
remarkable that the suggested approach allows us to obtain such
solutions for 3D Stokes and continuity equations. As earlier, we
consider the cases of linearly and exponentially varying
viscosity.

\subsubsection{Linearly varying viscosity}

The statement of the problem is absolutely analogous to the
previous case. In 3D space we consider a parallelepiped $\Omega$:
$0\leq x\leq x_{size},0\leq y\leq y_{size}, 0\leq z\leq z_{size}.$
We assume that $\eta =ax+by+cz+e. $ We will mark the exact
solution obtained in Section 3 as $v_{x,a}, v_{y,a}, v_{z,a},
P_a$. Due to the uniqueness theorem, it is the solution of the
boundary problem in the parallelepiped $\Omega$ with the following
conditions at the boundary $\partial \Omega = \{x=0, x=x_{size},
y=0, y=y_{size},  z=0, z=z_{size}\}:$
$$
v_{x}\vert_{\partial \Omega} = v_{x,a}, v_{y}\vert_{\partial
\Omega} = v_{y,a}, v_{z}\vert_{\partial \Omega} = v_{z,a}.
$$


Let us compute the velocity and pressure with using of chosen
finite-difference algorithm. The corresponding numerical solution
is marked as $v_{x,n}, v_{y,n}, v_{z,n}, P_n$. The deviation of
these values from the exact solution ($v_{x,n}-v_{x,a}, v_{y,n}-
v_{y,a},  v_{z,n}- v_{z,a}, P_n - P_a$) is related with the error
of the numerical scheme. As in 2D case, we consider three error
norms: $L_{\infty}, L_1 , L_2$. We deal with cases of low and high
viscosity contrasts. Coefficients $a,b,c,e$ in the viscosity
formula are determined through given viscosity values ($\eta_1 =1,
\eta_2, \eta_3, \eta_4$) at four adjacent parallelepiped vertices.
We choose the following system parameters:

$$e=\eta_1,
    a=(\eta_3-\eta_1)/x_{size},
  $$
  $$
    b=(\eta_2-\eta_1)/y_{size},
      c=(\eta_4-\eta_1)/z_{size}.
$$
$$
\eta= a x + b y + c z +e,
$$
$$
\rho = \beta_1 \eta + \beta_2
$$
$$
 x_{size} = y_{size}= z_{size} = 1,
$$
$$
 G_x = 10, G_y = 10,G_z = 0,
$$
$$
\eta_1 =1, \beta_1 = 1, \beta_2 = 3\times 10^3,
$$
Results are presented at Fig. 13-20. Figures 13-16 correspond to
the case of low viscosity contrast, figures 17-20 - to the case of
high viscosity contrast. Figures 13, 14 and figures 17, 18 shows
the velocity components and the pressure distribution for the
central cross-section of the parallelepiped. The analytical and
the numerical solutions and also the error (the solutions
deviations) are shown. Fig. 15, 19 characterizes the distribution
of the viscosity and the density for the same cross-section. In
Fig. 16, 20 the errors for a sequence of grid resolution used to
solve the  equations numerically are provided. Positive slope of
the curves shows the algorithm convergence. It should be mentioned
that the numerical procedure for 3D case takes essentially greater
time than in 2D case. Qualitatively, the results are similar to
that for  the corresponding 2D case. The numerical procedure works
better for low viscosity contrast.

\begin{figure}
\centerline{\includegraphics[width=3in]{fig_1a_low_lin.jpg}}
 \caption{ Distribution of $v_x$ and $v_y$; 3D case,
 linearly varying viscosity, low viscosity contrast
 ($\eta_2 = \eta_3 =  \eta_4 = 5$). }\label{fig_1a_low_lin}
\end{figure}

\begin{figure}
\centerline{\includegraphics[width=3in]{fig_1b_low_lin.jpg}}
 \caption{ Distribution of $v_z$ and $P$; 3D case,
 linearly varying viscosity, low viscosity contrast
 ($\eta_2 = \eta_3 =\eta_4 = 5$). }\label{fig_1a_low_lin}
\end{figure}



\begin{figure}
\centerline{\includegraphics[width=3in]{fig_4_low_lin.jpg}}
 \caption{ Distribution of viscosity $\eta$ and
 density $\rho$; 3D case,
 linearly varying viscosity, low viscosity contrast
 ($\eta_2 = \eta_3 =\eta_4 = 5$). }\label{fig_4_low_lin}
\end{figure}

\begin{figure}
\centerline{\includegraphics[width=3in]{fig_3_low_lin.jpg}}
 \caption{ Logarithm of the relative error via
 logarithm of the grid step; 3D case,
 linearly varying viscosity, low viscosity contrast
 ($\eta_2 = \eta_3 =\eta_4 = 5$); blue line - pressure, red - $v_x$,
 black - $v_y$, green - $v_z$; line - $L_1$-error,
 dashed line - $L_{\infty}$-error, line with dots -
 $L_2$-error. }\label{fig_3_low_lin}
\end{figure}


\begin{figure}
\centerline{\includegraphics[width=3in]{fig_1a_high_lin.jpg}}
 \caption{ Distribution of $v_x$ and $v_y$; 3D case,
 linearly varying viscosity, high viscosity contrast
 ($\eta_2 = \eta_3 =\eta_4 = 100$). }\label{fig_1a_high_lin}
\end{figure}

\begin{figure}
\centerline{\includegraphics[width=3in]{fig_1b_high_lin.jpg}}
 \caption{ Distribution of $v_z$ and $P$; 3D case,
 linearly varying viscosity, high viscosity contrast
 ($\eta_2 = \eta_3 =\eta_4 = 100$). }\label{fig_1a_high_lin}
\end{figure}



\begin{figure}
\centerline{\includegraphics[width=3in]{fig_4_high_lin.jpg}}
 \caption{ Distribution of viscosity $\eta$ and
 density $\rho$; 3D case,
 linearly varying viscosity, high viscosity contrast
 ($\eta_2 = \eta_3 =\eta_4 = 100$).  }\label{fig_4_high_lin}
\end{figure}

\begin{figure}
\centerline{\includegraphics[width=3in]{fig_3_high_lin.jpg}}
 \caption{ Logarithm of the relative error via
 logarithm of the grid step; 3D case,
 linearly varying viscosity, high viscosity contrast
 ($\eta_2 = \eta_3 =\eta_4 = 100$); blue line - pressure, red - $v_x$,
 black - $v_y$, green - $v_z$; line - $L_1$-error,
 dashed line - $L_{\infty}$-error, line with dots -
 $L_2$-error. }\label{fig_3_high_lin}
\end{figure}







\subsubsection{Exponentially varying viscosity}

As in 2D case, we consider also exponentially varying viscosity.
Consideration is absolutely parallel to that in the previous
section. To have a possibility of comparison we take here the same
system parameters. Naturally, the viscosity and, correspondingly,
density distributions are now exponential. The system parameters
are chosen by the following manner:

$$C=\eta_1,
    a=(\log(\eta_3)-\log(\eta_1))/x_{size},
  $$
  $$
    b=(\log(\eta_2)-\log(\eta_1))/y_{size},
      c=(\log(\eta_4)-\log(\eta_1))/z_{size}.
$$
$$
\eta= C \exp{(a x + b y + c z)},
$$
$$
\rho = \beta_1 \eta + \beta_2
$$
$$
 x_{size} = y_{size}= z_{size} = 1,
$$
$$
 G_x = 10, G_y = 10,G_z = 0,
$$
$$
\eta_1 =1, \beta_1 = 1, \beta_2 = 3\times 10^3,
$$
The results are presented at Fig. 21-28. Figures 21-24 correspond
to the case of low viscosity contrast, figures 25-28 - to the case
of high viscosity contrast. It occurs that for exponentially
varying viscosity the numerical algorithm works better than for
linearly varying viscosity (compare the velocity and the pressure
distributions and the corresponding errors at Fig. 21, 22, 25, 26
with Fig. 13, 14, 17, 18). It is in correlation with 2D case. The
algorithm convergence, i.e. the dependence of the errors on the
grid resolution is shown at Fig. 24, 28. One can see that
"exponential" case gives us better convergence than the
corresponding "linear" case.


\begin{figure}
\centerline{\includegraphics[width=3in]{fig_1a_3D_low.jpg}}
 \caption{ Distribution of $v_x$ and $v_y$; 3D case,
 exponentially varying viscosity, low viscosity contrast
 ($\eta_2 = \eta_3 = \eta_4 = 5$). }\label{fig_1a_3D_low}
\end{figure}

\begin{figure}
\centerline{\includegraphics[width=3in]{fig_1b_3D_low.jpg}}
 \caption{ Distribution of $v_z$ and $P$; 3D case,
 exponentially varying viscosity, low viscosity contrast
 ($\eta_2 = \eta_3 = \eta_4 = 5$). }\label{fig_1b_3D_low}
\end{figure}



\begin{figure}
\centerline{\includegraphics[width=3in]{fig_4_3D_low.jpg}}
 \caption{ Distribution of viscosity $\eta$ and
 density $\rho$; 3D case,
 exponentially varying viscosity, low viscosity contrast
 ($\eta_2 = \eta_3 = \eta_4 = 5$). }\label{fig_4_3D_low}
\end{figure}

\begin{figure}
\centerline{\includegraphics[width=3in]{fig_3_3D_low.jpg}}
 \caption{ Logarithm of the relative error via
 logarithm of the grid step; 3D case,
 exponentially varying viscosity, low viscosity contrast
 ($\eta_2 = \eta_3 = \eta_4 = 5$); blue line - pressure, red - $v_x$,
 black - $v_y$, green - $v_z$; line - $L_1$-error,
 dashed line - $L_{\infty}$-error, line with dots -
 $L_2$-error. }\label{fig_3_3D_low}
\end{figure}


\begin{figure}
\centerline{\includegraphics[width=3in]{fig_1a_high.jpg}}
 \caption{ Distribution of $v_x$ and $v_y$; 3D case,
 exponentially varying viscosity, high viscosity contrast
 ($\eta_2 = \eta_3 = \eta_4 = 100$). }\label{fig_1a_high}
\end{figure}

\begin{figure}
\centerline{\includegraphics[width=3in]{fig_1b_high.jpg}}
 \caption{ Distribution of $v_z$ and $P$; 3D case,
 exponentially varying viscosity, high viscosity contrast
 ($\eta_2 = \eta_3 = \eta_4 = 100$). }\label{fig_1b_high}
\end{figure}

\begin{figure}
\centerline{\includegraphics[width=3in]{fig_4_high.jpg}}
 \caption{ Distribution of viscosity $\eta$ and
 exponentially varying viscosity, high viscosity contrast
 ($\eta_2 = \eta_3 = \eta_4 = 100$).  }\label{fig_4_high}
\end{figure}

\begin{figure}
\centerline{\includegraphics[width=3in]{fig_3_high.jpg}}
 \caption{ Logarithm of the relative error via
 logarithm of the grid step; 3D case,
 exponentially varying viscosity, high viscosity contrast
 ($\eta_2 = \eta_3 = \eta_4 = 100$); blue line - pressure, red - $v_x$,
 black - $v_y$, green - $v_z$; line - $L_1$-error,
 dashed line - $L_{\infty}$-error, line with dots -
 $L_2$-error. }\label{fig_3_high}
\end{figure}


%\appendixtables

\section{Error decreasing via the grid resolution decreasing}

The dependence of the convergence on the viscosity contrast is
shown in the following tables. Namely, the viscosities $\eta_2,
\eta_3$ run through a set of values from $2$ to $10000$.
 For each pair of $\eta_2, \eta_3$ we perform the benchmarking
 procedure described above and obtain the curve describing the
 dependence of the logarithm of the error norm on the logarithm of
 the grid resolution. The tangent of the slope angle of this curve
 gives us an input of the table. Full set of tables is given in
 supplementary materials. Below, we show several examples of such
 tables.

\subsection{Linearly varying viscosity}

Tables~A1--A9 contain log rates of error decreasing via the grid
resolution decreasing  (the curve slope at Figs. 3 and~6 in
logarithmic scale) for different viscosity contrasts in the case
of
linearly varying viscosity. Calculations were made for the
following system parameters: \hack{\allowdisplaybreaks}
\begin{align}
&\nonumber x_{\text{size}} = y_{\text{size}} = 1,\quad G_x =
0,\quad G_y = 10,
\\
& \nonumber \eta_1 =1,\quad \beta_1 = 10^2,\quad \beta_2 = 3\times
10^3,
\\
& \nonumber c=\eta_1,\quad a=(\eta_3 -
\eta_1)/x_{\text{size}},\quad b=(\eta_2 -\eta_1)/y_{\text{size}},
\\
& \nonumber \rho = \beta_1 (ax + by +c) + \beta_2.
\end{align}
Empty table cells correspond to the case when we have very small
errors for all considered grid steps.


\subsection{Exponentially varying viscosity}

Tables~A10--A18 contain log rates of error decreasing with
decreasing grid step (the curve slope at Figs.~9 and 12 in
logarithmic scale) decreasing for different viscosity contrasts in
the case of exponentially varying viscosity. System parameters are
the same as in the case of linearly varying viscosity with natural
changes: \hack{\allowdisplaybreaks}
\begin{align}
&\nonumber c=\eta_1,\quad a=(\log\eta_3 -
\log\eta_1)/x_{\text{size}},\quad b=(\log\eta_2
-\log\eta_1)/y_{\text{size}},
\\
& \nonumber \rho = \beta_1 c \exp{(ax + by)} + \beta_2.
\end{align}

%\Supplementary{zip}





\begin{table}
\caption{Pressure;  $L_{\infty}$-error.}
\begin{tabular}{lccccccc}
\tophline
\multicolumn{1}{r}{$\eta_2$}&2&5&20&100&300&1000&10\,000\\
$\eta_3$&&&&&&&\\
\middlehline
2&0.89&0.97&0.81&0.46&0.11&$-$0.19&$-$0.40\\
5&0.96&0.93&0.89&0.77&0.14&$-$0.14&$-$0.39\\
20&0.83&0.89&0.86&0.77&0.73&$-$0.06&$-$0.33\\
100&0.80&0.79&0.77&0.83&0.77&0.46&$-$0.21\\
300&0.76&0.76&0.77&0.79&0.83&0.79&$-$0.11\\
1000&0.49&0.61&0.73&0.77&0.77&0.83&0.14\\
10\,000&$-$0.40&$-$0.39&$-$0.33&$-$0.21&0.13&0.41&0.80\\
\bottomhline
\end{tabular}
\end{table}

\begin{table}
\caption{Pressure;  $L_1$-error.}
\begin{tabular}{lccccccc}
\tophline
\multicolumn{1}{r}{$\eta_2$}&2&5&20&100&300&1000&10\,000\\
$\eta_3$&&&&&&&\\
\middlehline
2&1.40&0.96&1.16&0.74&0.41&0.04&$-$0.24\\
5&0.96&1.41&0.86&1.30&0.41&0.17&$-$0.20\\
20&1.16&0.86&1.37&1.21&0.60&0.24&$-$0.06\\
100&1.07&1.13&1.19&1.37&1.34&0.31&0.13\\
300&1.29&1.31&1.34&1.36&1.37&1.04&0.19\\
1000&0.37&0.51&0.77&1.16&1.33&1.36&0.20\\
10\,000&$-$0.30&$-$0.27&$-$0.14&0.04&0.14&0.27&1.27\\
\bottomhline
\end{tabular}
\end{table}

\begin{table}
\caption{Pressure;  $L_2$-error.}
\begin{tabular}{lccccccc}
\tophline
\multicolumn{1}{r}{$\eta_2$}&2&5&20&100&300&1000&10\,000\\
$\eta_3$&&&&&&&\\
\middlehline
2&1.42&1.14&1.24&0.86&0.46&0.08&$-$0.20\\
5&1.13&1.44&1.03&1.30&0.57&0.23&$-$0.16\\
20&1.23&1.03&1.39&1.27&0.83&0.43&$-$0.01\\
100&1.14&1.21&1.26&1.36&1.33&0.50&0.26\\
300&1.27&1.27&1.29&1.33&1.34&1.17&0.40\\
1000&0.56&0.70&1.03&1.24&1.31&1.36&0.39\\
10\,000&$-$0.21&$-$0.19&$-$0.04&0.23&0.34&0.43&1.27\\
\bottomhline
\end{tabular}
\end{table}

\begin{table}
\caption{Velocity $v_x$;  $L_{\infty}$-error.}
\begin{tabular}{lccccccc}
\tophline
\multicolumn{1}{r}{$\eta_2$}&2&5&20&100&300&1000&10\,000\\
$\eta_3$&&&&&&&\\
\middlehline
2&1.67&1.67&1.41&0.77&0.34&$-$0.03&$-$0.30\\
5&1.63&1.44&1.23&1.04&0.36&0.06&$-$0.26\\
20&1.50&1.39&1.11&0.83&0.26&0.09&$-$0.17\\
100&0.59&0.79&0.73&1.08&1.43&0.06&$-$0.04\\
300&1.59&1.59&1.36&1.36&1.41&0.06&$-$0.01\\
1000&$-$0.29&$-$0.13&0.29&0.60&1.41&0.03&$-$0.01\\
10\,000&$-$0.67&$-$0.64&$-$0.51&$-$0.21&0.07&0.03&0.00\\
\bottomhline
\end{tabular}
\end{table}

\begin{table}
\caption{Velocity $v_x$;  $L_1$-error.}
\begin{tabular}{lccccccc}
\tophline
\multicolumn{1}{r}{$\eta_2$}&2&5&20&100&300&1000&10\,000\\
$\eta_3$&&&&&&&\\
\middlehline
2&1.68&2.01&1.91&1.40&0.83&0.40&0.13\\
5&1.51&1.75&1.96&1.93&1.09&0.60&0.16\\
20&1.44&1.47&1.70&2.00&1.50&0.99&0.31\\
100&1.17&1.43&1.56&1.85&2.09&1.39&0.70\\
300&1.63&1.61&1.63&1.66&1.89&1.87&1.04\\
1000&0.64&0.84&1.24&1.59&1.61&1.36&1.29\\
10\,000&$-$0.06&$-$0.01&0.13&0.53&0.84&1.09&1/12\\
\bottomhline
\end{tabular}
\end{table}

\begin{table}
\caption{Velocity $v_x$;  $L_2$-error.}
\begin{tabular}{lccccccc}
\tophline
\multicolumn{1}{r}{$\eta_2$}&2&5&20&100&300&1000&10\,000\\
$\eta_3$&&&&&&&\\
\middlehline
2&1.81&1.99&1.81&1.20&0.70&0.29&0.01\\
5&1.53&1.61&1.80&1.79&0.91&0.47&0.04\\
20&1.47&1.54&1.61&1.81&1.09&0.77&0.20\\
100&1.13&1.39&1.54&1.69&2.04&0.94&0.53\\
300&1.64&1.63&1.63&1.61&1.70&1.49&0.77\\
1000&0.56&0.77&1.23&1.57&1.60&1.15&0.90\\
10\,000&$-$0.19&$-$0.13&0.01&0.41&0.69&0.96&0.88\\
\bottomhline
\end{tabular}
\end{table}

\begin{table}
\caption{Velocity $v_y$;  $L_{\infty}$-error.}
\begin{tabular}{lccccccc}
\tophline
\multicolumn{1}{r}{$\eta_2$}&2&5&20&100&300&1000&10\,000\\
$\eta_3$&&&&&&&\\
\middlehline
2&1.75&1.63&1.47&0.64&0.03&$-$0.40&$-$0.67\\
5&1.67&1.44&1.37&1.23&0.24&$-$0.26&$-$0.64\\
20&1.40&1.23&0.97&0.83&0.34&0.09&$-$0.51\\
100&0.83&0.79&0.79&1.33&1.56&0.07&$-$0.21\\
300&1.66&1.76&1.91&1.91&1.91&0.66&0.04\\
1000&$-$0.01&0.06&0.04&0.37&0.86&1.17&0.03\\
10\,000&$-$0.30&$-$0.27&$-$0.16&$-$0.06&$-$0.03&$-$0.03&$-$0.30\\
\bottomhline
\end{tabular}
\end{table}

\begin{table}
\caption{Velocity $v_y$;  $L_1$-error.}
\begin{tabular}{lccccccc}
\tophline
\multicolumn{1}{r}{$\eta_2$}&2&5&20&100&300&1000&10\,000\\
$\eta_3$&&&&&&&\\
\middlehline
2&1.98&1.51&1.44&1.17&0.59&0.16&$-$0.13\\
5&2.01&1.90&1.47&1.63&0.84&0.34&$-$0.10\\
20&1.90&1.94&1.94&1.57&1.30&0.73&0.06\\
100&1.51&1.76&2.00&2.06&1.66&0.81&0.43\\
300&1.87&1.96&2.04&2.09&2.07&1.44&0.76\\
1000&0.66&0.86&1.30&1.81&2.07&2.07&0.93\\
10\,000&0.14&0.19&0.34&0.74&1.07&1.34&1.87\\
\bottomhline
\end{tabular}
\end{table}

\begin{table}
\caption{Velocity $v_y$;  $L_2$-error.}
\begin{tabular}{lccccccc}
\tophline
\multicolumn{1}{r}{$\eta_2$}&2&5&20&100&300&1000&10\,000\\
$\eta_3$&&&&&&&\\
\middlehline
2&1.99&1.53&1.49&1.14&0.51&0.07&$-$0.21\\
5&1.99&1.90&1.54&1.61&0.79&0.26&$-$0.19\\
20&1.81&1.80&1.79&1.56&1.24&0.63&$-$0.03\\
100&1.27&1.47&1.79&2.03&1.61&0.94&0.34\\
300&1.86&1.93&1.99&2.04&2.04&1.43&0.60\\
1000&0.37&0.53&0.90&1.56&1.97&2.03&0.74\\
10\,000&0.019&0.04&0.20&0.54&0.76&0.89&1.23\\
\bottomhline
\end{tabular}
\end{table}


\begin{table}
\caption{Pressure;  $L_{\infty}$-error.}
\begin{tabular}{lccccccc}
\tophline
\multicolumn{1}{r}{$\eta_2$}&2&5&20&100&300&1000&10\,000\\
$\eta_3$&&&&&&&\\
\middlehline
2&0.94&0.83&0.67&1.01&1.24&1.41&0.84\\
5&0.87&0.64&0.71&0.84&1.24&1.31&1.16\\
20&0.73&0.83&0.81&1.01&1.29&1.19&1.04\\
100&0.80&0.89&1.07&1.30&1.23&1.14&0.96\\
300&0.90&0.99&1.13&1.23&1.19&1.09&0.91\\
1000&1.04&1.13&1.14&1.11&1.09&1.03&0.86\\
10\,000&1.19&1.13&0.70&0.91&0.87&0.84&0.71\\
\bottomhline
\end{tabular}
\end{table}

\begin{table}
\caption{Pressure;  $L_1$-error.}
\begin{tabular}{lccccccc}
\tophline
\multicolumn{1}{r}{$\eta_2$}&2&5&20&100&300&1000&10\,000\\
$\eta_3$&&&&&&&\\
\middlehline
2&1.40&0.89&1.10&1.40&1.46&1.44&1.37\\
5&0.89&1.13&1.17&1.34&1.41&1.43&1.36\\
20&0.97&1.14&1.31&1.40&1.40&1.40&1.33\\
100&1.20&1.36&1.31&1.39&1.39&1.36&1.26\\
300&1.37&1.36&1.27&1.34&1.36&1.31&1.21\\
1000&1.39&1.36&1.27&1.29&1.30&1.26&1.17\\
10\,000&1.33&1.29&1.23&1.17&1.14&1.13&1.06\\
\bottomhline
\end{tabular}
\end{table}

\begin{table}
\caption{Pressure;  $L_2$-error.}
\begin{tabular}{lccccccc}
\tophline
\multicolumn{1}{r}{$\eta_2$}&2&5&20&100&300&1000&10\,000\\
$\eta_3$&&&&&&&\\
\middlehline
2&1.46&1.03&1.20&1.47&1.56&1.53&1.40\\
5&1.03&1.13&1.23&1.39&1.53&1.51&1.39\\
20&1.13&1.24&1.31&1.46&1.51&1.46&1.31\\
100&1.27&1.39&1.39&1.46&1.43&1.37&1.23\\
300&1.37&1.41&1.39&1.40&1.37&1.31&1.19\\
1000&1.41&1.41&1.36&1.33&1.30&1.24&1.11\\
10\,000&1.37&1.34&1.26&1.17&1.14&1.11&1.00\\
\bottomhline
\end{tabular}
\end{table}

\begin{table}
\caption{Velocity $v_x$;  $L_{\infty}$-error.}
\begin{tabular}{lccccccc}
\tophline
\multicolumn{1}{r}{$\eta_2$}&2&5&20&100&300&1000&10\,000\\
$\eta_3$&&&&&&&\\
\middlehline
2&1.72&1.71&1.61&1.81&1.74&1.70&1.63\\
5&1.69&1.65&1.41&1.83&1.73&1.66&1.56\\
20&1.66&1.59&1.36&1.80&1.70&1.56&1.44\\
100&1.71&1.79&1.90&1.76&1.59&1.46&1.36\\
300&1.87&2.00&1.89&1.71&1.56&1.46&1.31\\
1000&1.79&1.87&1.93&1.71&1.54&1.41&1.26\\
10\,000&1.49&1.66&1.91&1.67&1.51&1.37&1.17\\
\bottomhline
\end{tabular}
\end{table}

\begin{table}
\caption{Velocity $v_x$;  $L_1$-error.}
\begin{tabular}{lccccccc}
\tophline
\multicolumn{1}{r}{$\eta_2$}&2&5&20&100&300&1000&10\,000\\
$\eta_3$&&&&&&&\\
\middlehline
2&1.89&2.03&2.06&2.17&2.10&2.06&2.01\\
5&1.61&1.78&2.06&2.24&2.14&2.10&2.01\\
20&1.59&1.83&2.21&2.21&2.16&2.13&2.04\\
100&1.64&1.87&2.10&2.21&2.14&2.10&2.04\\
300&1.76&1.91&2.04&2.21&2.14&2.07&2.01\\
1000&1.84&1.94&2.04&2.13&2.14&2.06&1.97\\
10\,000&1.77&1.87&1.96&2.00&2.01&2.00&1.91\\
\bottomhline
\end{tabular}
\end{table}

\begin{table}
\caption{Velocity $v_x$;  $L_2$-error.}
\begin{tabular}{lccccccc}
\tophline
\multicolumn{1}{r}{$\eta_2$}&2&5&20&100&300&1000&10\,000\\
$\eta_3$&&&&&&&\\
\middlehline
2&1.92&2.00&1.97&2.03&2.00&1.96&1.91\\
5&1.66&1.82&1.93&2.09&2.01&1.97&1.90\\
20&1.63&1.84&2.06&2.09&2.01&1.96&1.87\\
100&1.71&1.93&2.06&2.04&1.97&1.90&1.83\\
300&1.87&1.99&2.03&2.00&1.93&1.84&1.74\\
1000&1.93&2.00&2.03&2.00&1.93&1.84&1.74\\
10\,000&1.76&1.89&1.96&1.94&1.90&1.83&1.67\\
\bottomhline
\end{tabular}
\end{table}

\begin{table}
\caption{Velocity $v_y$;  $L_{\infty}$-error.}
\begin{tabular}{lccccccc}
\tophline
\multicolumn{1}{r}{$\eta_2$}&2&5&20&100&300&1000&10\,000\\
$\eta_3$&&&&&&&\\
\middlehline
2&1.76&1.67&1.83&1.87&1.71&1.71&1.56\\
5&1.73&1.56&1.53&2.09&1.99&1.90&1.89\\
20&1.67&1.46&1.37&1.96&1.89&1.86&1.84\\
100&1.57&1.27&1.80&1.77&1.69&1.64&1.61\\
300&1.27&1.80&1.79&1.67&1.57&1.53&1.50\\
1000&1.81&1.81&1.73&1.56&1.49&1.43&1.36\\
10\,000&1.71&1.16&1.56&1.40&1.36&1.29&1.19\\
\bottomhline
\end{tabular}
\end{table}

\begin{table}
\caption{Velocity $v_y$;  $L_1$-error.}
\begin{tabular}{lccccccc}
\tophline
\multicolumn{1}{r}{$\eta_2$}&2&5&20&100&300&1000&10\,000\\
$\eta_3$&&&&&&&\\
\middlehline
2&1.91&1.63&1.74&2.16&2.13&2.11&2.03\\
5&2.03&1.91&1.97&2.13&2.17&2.19&2.14\\
20&2.03&2.01&2.26&2.21&2.20&2.20&2.14\\
100&2.01&2.24&2.29&2.31&2.30&2.26&2.17\\
300&2.26&2.34&2.27&2.23&2.26&2.26&2.17\\
1000&2.24&2.24&2.23&2.17&2.17&2.17&2.16\\
10\,000&2.07&2.10&2.11&2.11&2.09&2.07&2.07\\
\bottomhline
\end{tabular}
\end{table}




\begin{table}
\caption{Velocity $v_y$;  $L_2$-error.}
\begin{tabular}{lccccccc}
\tophline
\multicolumn{1}{r}{$\eta_2$}&2&5&20&100&300&1000&10\,000\\
$\eta_3$&&&&&&&\\
\middlehline
2&1.97&1.66&1.81&2.10&2.01&1.94&1.83\\
5&2.00&1.93&1.99&2.11&2.10&2.04&1.94\\
20&1.97&1.90&2.09&2.14&2.11&2.07&2.00\\
100&1.89&1.96&2.11&2.09&2.06&2.04&1.97\\
300&2.03&2.11&2.09&2.01&1.97&1.96&1.93\\
1000&2.11&2.06&2.04&1.96&1.91&1.87&1.86\\
10\,000&1.96&1.94&1.93&1.87&1.83&1.77&1.71\\
\bottomhline
\end{tabular}
\end{table}



\end{document}
