My research was in numerical analysis, the design of efficient algorithms that solve partial differential equations.
In many problems that arise in nature we can measure quantities on a very fine scale but we are interested in an answer on a much larger scale. For example in underground flows through a rock we can measure the properties of the rock on the scale of centimeters, but we are interested in the transport of the fluid across kilometers. If we simulated the flow on the fine scale we would be wasting computational time. Instead we try to derive effective equations for the flow on the large scale and solve these equations instead. This process is called upscaling.
Homogenization is a mathematical theory that provides us with large scale equation from the fine scale equations. We consider a sequense of different problems each with more and more fluctuations on a smaller and smaller length scale. In the limit the fluctuations disappear and we obtain equations with homogeneous coefficients, which can be solved on the large scale. It is remarkable that such a mathematical procedure, perhaps counterintuitive to our physical intuition, can provide accurate models. Computing that limit can be very challenging.
And indeed we do get pretty pictures:
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Imagine that this rock is saturated with oil. To extract it we drill wells to the left and right of it and pump water in the left well. The water displaces the oil into the right well from where we can collect it. |
Simulation of two phase flow in a porous rock. Water (red) displaces the oil (blue) and breaks through. Most trasport of water and oiloccurs through fast channels in highly permeable regions. |
Powerpoint slides of my defense, full thesis in pdf form.
Prior to this I worked on a moving mesh for the evolution of a vortex sheet, on a moving mesh for ray tracing, on removing the stiffness from the immersed boundary method.
To write my thesis I adapted Ling Li's latex class to contain environments relevant to mathematicians. You can download the latex class and a lyx layout file and the caltech logo in eps and pdf form. For directions on how to use them you should visit Ling Li's homepage.
