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-\documentclass[12pt]{article}
-\usepackage[dvips]{graphicx}
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-\begin{document}
-\section{Result}
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-For the test case in Introduction,we do numerical computation on four meshes:
-$$h_{1}=2^{-3}, h_{2}=2^{-4}, h_{3}=2^{-5}, h_{4}=2^{-6}$$
-
-Our permeability is $K(x,y)=\frac{1.0}{2+1.99*sin(2 \pi \frac{2x-y}{\epsilon})}$:\\
-
-\includegraphics[width=6 in]{perm.eps}
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-Because of heterogeneity in the medium, the solution for velocity and saturatio are periodic.
-Compare the following pictures on four meshes at time $t=T$:\\
-Velocity-x:\\
-\includegraphics[width=5 in]{solu14meshes.eps}
-\\
-Velocity-y:\\
-\includegraphics[width=5 in]{solu24meshes.eps}
-\\
-Pressure:\\
-\includegraphics[width=5 in]{solp4meshes.eps}
-\\
-Saturation:\\
-\includegraphics[width=5 in]{sols4meshes.eps}
-\\
-From above comparation, we can see pressure is stable but velocity and saturation are not.
-The reason is heterogeneity of the medium and some complexity fo the dynamic systems.
-By our direct numerical computation, only fine mesh solution is able to catch the subgrid properties.
-That means an accurate well-posed computation requires tremendous amount of computer memory and CPU time.
-But usually ,it easily exceed the limit of today's computer resources.\\
-There are some alternative approaches have been developed. A common approach is to "scale up" a heterogeneous medium.
-This method is to find an effective representation of permeablility on a coarse mesh so that the large scale flow can be correctly computed on this mesh.The computational cost is thus greatly reduced.
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-At last,let's see Oil Production Rate on the boundary $\Gamma_{2}$:
-$$ PR(t)=1-\frac{\int_{\Gamma_{2}} (\mathbf{u}\cdot \mathbf{n})F(S)dx}{\int_{\Gamma_{2}} (\mathbf{u} \cdot \mathbf{n})dx}$$
-
-\includegraphics[width=6 in]{pr4meshes.eps}
-
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-\end{document}
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