Mainly the issue really boils down the the Gemara in Shabat chapter 3 about cooking with תולדות חמה or in חמי טבריה. [Heat generated by some derivative of solar energy, not fire]. The relevant sources are the Chazon Ish, and the book of one of his disciples that disagreed with the Chazon Ish, and the Gemaras from where the Chazon Ish derives his law from. They are the gemaras about putting a bed or a candelabra that are made out of parts together. I was back at the Mir in NY for a sort time and looked at the Chazon Ish and was impressed. I then asked Rav Nelkenbaum about it and he said an אדם גדול told him the essay of the Chazon Ish on this subject is simply and plainly wrong.
That is the sum total of the relevant information I have about this subject. The only thing I might add is the argument between the Rambam and Raavad about a vessel that needs to be put together to be operative in laws of טומאה וטהרה but after thinking that over I did not think it was relevant. In any case the arguments that forbid electricity on Shabat or cooking with electricity are amazingly flimsy and concocted out of thin air.
As they say in Israel "If you want to be frum, (extra strict) then do it on your own חשבון (expense). Do not force it down the throats of others."