How can I make sure my evening project code is mine?

Question

I'm a physicist with a CS degree and just started my PhD at a tech company (wanted to do applied research). It deals with large scale finite element simulations.

After reviewing their current approach, I think that a radically different method has to be applied (they are using a commercial tool which is very limited).

I'd rather base my research on an open source finite element solver and write a program which makes use of it. I'd like to develop this idea in the evenings, because that's the time that best suits me for programming (during the day I prefer reading and maths) and use it at a late stage of my PhD.

I'd like to have the option to release my program as open source on my website as a reference, for future personal or even commercial (e.g. consulting) use.

How can I make sure that my company doesn't claim the code ownership?

I thought that a version control system could help (check out only in the evening). This would document that I programmed not during regular office hours (documented elsewhere). But these data can be easily manufactured. Any other ideas?

I want to stress that I'm not interested in selling software and neither is my company.

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Very interesting responses so far. This clearly helps me. Some remarks:

• I'm not restrained by my work contract. National law says that the company owns anything I produce during working hours and no special agreement has been made (my employer is not selling software and may be a bit naive on this side). They mostly use software and non of my colleagues is a serious programmer.
• Secondly, I need to rethink the point raised by @Mark about trade secrets. This is quite serious in the particular industry.
• Thirdly, I care a lot about no to upset my supervisor/ boss. But, and this is the motivation for this question, I'd like to keep the innovative part of my work a bit separated so I can reuse it or at least demonstrate it as a reference work.