Euclid's Algorithm [1]; it's one of the oldest algorithms still in use, simple, elegant, powerful, has lots of generalizations, many theoretical and practical applications...
by beatiful, I meant an algorithm that it is simple, easy to understand and still has an intelligent solution. I'm not sure that algorithm in computiational fluid dynamics feat...
Those terms, simple and easy, should be taken as relative to the system being described.
I would argue that CFD models are surprisingly simple relative to the task of understanding multi-phase, particle-laden, turbulent fluids.
Read only if you're up for a good laugh: ahem... Bubble Sort!! ;-)
Not sure about most beautiful, but Union-Find definitely is very clever, clean, and useful.
I really like AVL trees (or balanced trees) and the way they are restructured.