For simplicity, I'll refer to all polystyrene foam, both extruded and expanded, as "Styrofoam"; the adhesives will be the same. There are a lot of adhesives that will stick Styrofoam together. Most of those will be adequate for an art project where nothing will be under stress. If the glue joint needs to have mechanical strength similar to the Styrofoam, itself, (i.e., you need to "weld" the pieces together to have strength similar to having originally molded a single piece), only some of the adhesives will do that.
The joint strength will be affected by several things:
- The surface area of the joint. If there is a lot of surface area, even an inexpensive, weak glue, like PVA craft glue, will have a lot of holding strength if you give it time to dry.
- Drying time. For adhesives that cure by evaporation, a large surface area can take a very long time for the glue to cure. The glue joint is sealed in an air-tight "container" between the pieces, and evaporation happens at the exposed edge after migrating through the glue. It can take a week or longer for the glue joint to have real strength. You will have a strong joint only if you have the time to wait.
Surface texture. Some Styrofoam has a very smooth surface and some has a lot of surface texture. If there is significant surface texture, only the high points will be in contact; so it might be a large area, but not much of it is actually touching. OTOH, surface texture provides more surface for the adhesive to grab. That difference affects the type of adhesive that will work best.
For example, regular cyanoacrylates (super glue, the thin stuff), has the most strength as a microscopic layer between two tightly-mating surfaces. Globs of it filling voids doesn't have great strength. That will work on Styrofoam with a very smooth surface. On a textured surface, it will be an expensive adhesive that doesn't buy you much over something like PVA glue that fills the voids and has adequate time to cure.
Before getting to specific recommendations, I'll preserve here something I mentioned in a comment. If a water-based glue, like PVA craft glue, will be adequate for your needs (and you have time for it to cure), tacky glue, like Aleene's Tacky Glue, will hold better than plain PVA craft glue.
There are several adhesives that haven't been mentioned yet, that are the strongest if you need a "structural" joint. These will create joints that are stronger than the Styrofoam, itself (if you try to separate the pieces, the Styrofoam will break in another spot before the glue joint separates).
There's an interesting test of adhesives joining Styrofoam in this YouTube video. I found the results surprising. Note, though, that the tests were performed after 48 hours. A number of products failed because they hadn't cured yet. They may well have fared better if given sufficient time:
- Glidden Gripper Primer and Sealer. This is one I wouldn't have thought of, but it is apparently a well-known secret among crafters. It's inexpensive compared to many of the adhesives, and basically welds Styrofoam together. In the side-by-side test I linked to above, it came out the best.
- Gorilla Glue. It requires moisture for the glue to work. On large Styrofoam surfaces, curing time can be long if you rely on room humidity to seep into the joint. Mist one of the surfaces slightly before gluing to speed things up. If you need to avoid the glue expanding out of the joint, there's a non-foaming version. Otherwise, just apply it sparingly and not too close to the edge, and wipe any that oozes out before it hardens.
- Great Stuff (and similar) foam sealant. Use a tiny amount because it expands -- a few dots or a thin bead -- and press the parts together before it starts to harden. It may expand out the sides of the joint, so there can be some cleanup if you use too much. Wipe off excess while it's still liquid.
There are a number of other adhesives commonly used, including construction adhesive designed for the purpose and spray adhesive, but they aren't as strong as these three.