I have had a small amount of success implementing GPGPU programs using webGL. If you code your solution to webGL you are able to offload OS/driver stack concerns to the web browser implementers.
Here are two examples I've done. The first is matrix multiplication on the GPU. The second is a cellular automaton like simulation (the falling sand game popular a few years ago). Each incremental game state is calculated in a shader program and then used as input for the next increment.
Below I've linked github urls and two blog posts that were done about them. Unfortunately right now webGL is kind of a moving target and I doubt these examples still work out of the box. However, if you are interested in GPGPU on webGL I would encourage you to get involved as I think there are some tweaks to the standard that would make GPGPU life easier.
This could be very useful for running certain algorithms in web apps...