OK, I see what you mean - that should work. It’s not a huge saving in mass though, you still need to bring all the fuel for acceleration, plus a non-negligible fraction of what you need for deceleration. And the less fuel you carry for deceleration, the faster the resupply ships have to be.
Behold, the Bussard Ramjet!
It probably won’t work, for reasons explained in the Wikipedia article. But it’s a clever idea!
Just my own.
Why would it have to stop? The fuel ships are pointed at Alpha Centauri, all accelerating up to some set speed that lets them still keep some useful amount of fuel in reserve. Figure this:
Fuel Ship 1: Leaves Year 1. Very slowly accelerates up to .1c, then coasts
Fuel Ship 2: Leaves year 2. Very slowly accelerates up to .09c
Fuel Ship 3: Leaves year 3. Very slowly accelerates up to .08c
Fuel Ship 4: Leaves year 4. Very slowly accelerates up to .07c
Fuel Ship 5: Leaves year 5. Very slowly accelerates up to .06c
Fuel Ship 6: Leaves year 6. Very slowly accelerates up to .05c
Fuel Ship 7: Leaves year 7. Very slowly accelerates up to .04c
Fuel Ship 8: Leaves year 8. Very slowly accelerates up to .03c
Fuel Ship 9: Leaves year 9. Very slowly accelerates up to .02c
Fuel Ship 10: Leaves year 10. Very slowly accelerates up to .01c
Colony Ship: Leaves Year 11. As it accelerates, faster than any of the fuel ships, it catches up to each of them and relieves them of any unburnt fuel. Planned carefully enough, the colony ship won’t have to “stop”, though it might have to lose some velocity to match each fuel ship. Rather than burn braking fuel, maybe it just launches a unbreakable diamond tether and reels the fuel ship in. In any case, by the time it catches Fuel Ship 1, the plan is for them both to be cruising at or near 0.1c and be halfway to the destination. Colony Ship tops up its tanks from FS1’s reserve, then begins the long breaking routine.
Of course, the only fuel likely to make any of this practical is anti-matter. Isaac Asimov wrote and essay called Sail On in which he calculated that a single ship could likely carry all it needed to make the trip (I don’t recall the numbers, but it was a fairly piddling quantity of anti-matter, combined with a few tons of hydrogen that could supply an extra kick, being pushed into fusion by the anti-matter).
Stasis fields and lithobraking…
“I just flew in from Alpha Centauri, and boy are my arms tired!”
This hits on a different if not terminal problem that many experts have noted on this whole idea. At speeds any significant fraction of C, the tinniest little speck of anything will just destroy the entire ship. The space shuttles have already had problems with that and they the shuttles only fly at about 17,000 mph which is only the tiniest fraction of the speed any such ship would need to go to reach the nearest star within even a few thousand years.
Humans are smarter than most in the ape family but we still have a very limited lifespan and both bodies and brains with significant needs that not incoincidentally are designed for life hear on human earth. I have tons of optimism for all technology including inter-solar system travel but inter-stellar travel just seems laughable at best and probably impossible. That may explain the dearth of reputable UFO sightings recently.
I don’t think it would work for the acceleration phase. The resupply ships would be moving slower than you, so you lose speed when you capture it with your tether.
It should work for the deceleration phase though, like Omphaloskeptic said.
Sorry, but anything in deep space flight that can be disrupted by Zoidberg is going to get a pass.
But a variation on the idea might; you should have read to the bottom. Using a magnetic field to gather fuel might not work - but using a magnetic sail to produce drag and slow the ship might.
Okay, that’s got me laughing out loud!
ETA: Of course, after you hit the planet at 0.1 C, it’s going need a millenium or more to return to being desirable real estate…
Not a lot of good when you’re trying to slow down.
Why not? Turbojets can use thrust deflectors to brake, why shouldn’t the ramjet? Or why can’t you tweak the field to pump the fuel into the other end of the ramjet so it’s just firing forward in the first place. Alternatively, consider the possibility that you’re using a fission pile as your prime mover, and all the ramscoop field is collecting is reaction mass, not fusion fuel. You can squirt that out in any direction you like. Indeed, just collecting the interstellar hydrogen would provide a little friction braking (just as it imposes an upper limit on the speed of the ramjet in the first place).
Possibly wrong nitpick, wouldn’t this bring you to zero speed relative to your starting point, not the target? Relative galactic motion and all that…
If you eject it forwards, some of it is might be slowed down by the interstellar medium in front of you enough for you to collect it and use it again, making braking all the more efficient.
Except for the tiny details of making a ramscoop field (don’t think we have any idea how to do that at the moment, or if it’s even possible).
But if this is a long term voyage, how is a fission reactor going to be maintained? Where are you going to get new fuel rods?
I should point out that, assuming your fuel has non-zero mass and that your maximum rate of acceleration/deceleration isn’t limited by your puny earth bodies, then you shouldn’t stop and turn around halfway. Because you’re carrying less fuel for the deceleration phase, you can decelerate more rapidly than you accelerated due to your lower mass. This allows you to keep accelerating past the halfway point, because you know you can slow down faster.
Exactly how far past halfway you go depends what proportion of your vessel’s mass is the fuel expended on your complete journey.
There I think we have to start doing the math, as to how much power we’re getting out of the pile, how much fuel we can take along with us, how long the voyage is going to take… I don’t know if there’s even a theoretical chance of, say, a round trip to Alpha Centauri on one load of fuel; IANARS. If we can get there and find that the system has an asteroid belt then things start to look up, as we can go prospecting for radioactives (using our back-up light-sail to run a solar furnace) and incidentally shovel up a shitload of dust or ice to use as reaction mass to start the journey back. 
That’s one potential problem with fission - even if you take spare fuel rods, they decay before you’ve had the chance to use them.
Well if it is taking 1000 years to get there, I would think that much faster ships invented 500, 600, 700 years after your departure would catch up with you and use ther tractor beams to slow you down
Sounds like a case for using a breeder reactor then - assuming there is some kind of rod we can use that is initially low-activity, slow-decay, and which we can bombard with neutrons shortly before use to make it high-activity, quick-decay. How are we fixed for converting U[sup]238[/sup] to plutonium, and then burning the plutonium? (Or converting U[sup]238[/sup] to U[sup]235[/sup].)
Wubba? Just what are you using in your piles?
[sup]235[/sup]U has a half-life of 7.038·10[sup]8[/sup] years - if your trip is going to last long enough that you’re going to signifigantly reduce the amount of [sup]235[/sup]U in your fuel cells, it’s time to re think things.
The more common [sup]238[/sup]U has a half-life of 4.46·10[sup]9[/sup] - ten times longer.
Yes, Pu has shorter half-lives, but I think that U would be the better fissile material.
I don’t think there’s any need to consider a breeder reactor, so you can reprocess your fuel. That’s just too much mass, IMNSHO, to carry along.