Why did Rhaegar take Lyanna to Dorne?

Question

Considering Rhaegar Targaryen was married to Elia Martell of Dorne, wasn't it rather stupid of him to take Lyanna Stark to the Tower of Joy, which is situated in the Red Mountains of Dorne?

Even considering the relatively free sexual norms in Dorne, and that Targaryens practiced polygamy, the Martell family might not have been too thrilled at Rhaegar treating Elia this way. As Oberyn states, it is a big insult to Elia's, and therefore the Martell family's honor. So not only would they have been angry at Lyanna's and Rhaegar's affair/marriage, wouldn't her presence in their land (or at their borders) not infuriate them even more?

I can think of a few explanations:

1. Rhaegar's and Lyanna's affair was secret and the Martells did not know that Lyanna was in Dorne.
2. The Martells believed that Rhaegar was "the Prince that was Promised" (as he believed himself at some point) and that he needed to have three children (Aegon, Rhaenys and Visenya, named after Aegon the Conqueror and his sisters), while Elia had given birth to two children before being rendered too weak, or possibly infertile, to have a third. In this case the Martells might have agreed with the alleged kidnapping/elopement. But in place of a little Visenya, Lyanna of course gave birth to Jon Targaryen-Snow.
3. This might have been a truly stupid decision on Rhaegar's part and GRRM will explain this later in the story.

Which of these is plausible (in-universe), or might there be a better explanation? What did I miss?

In case the TV series and the book series contradict each other, explanations from the book series are preferred!

Answer 1

Canon Answer

The canon answer is: We do not know yet. There is not much data available about what went on in Tower of Joy and Rhaegar's mind. All the people who might have known what happened there are dead except Howland Reed. And even Howland Reed can't be expected to know what was actually Rhaegar thinking. Only man who might have known it was Arthur Dayne, Rhaegar's friend. Arthur was killed at tower of joy along with other white swords posted there.

It is most plausibly going to be explained in the upcoming installments.

Speculation

There is no reason to assume that Prince Doran had any way of knowing that Rhaegar was in Dorne.

Even if he knew about it, It is not place of Prince Doran to tell Rhaegar what to do and what not to do. Rhaegar is a prince of the blood and heir to the throne of Aegon the conqueror. Example is Prince Aegon (Later King Aegon V). Lords positively hated him but none dared to tell the Prince to leave their lands. However the counter example here is that Lady Jeyne Arryn of Vale expelled Prince Daemon Targaryen from Vale and forbid his entry into her domains ever again. However the difference here is, Daemon was younger brother of the King with no chance of inheriting the throne and no lands of his own so it was easier to offend him. Rhaegar was however the crown prince and Prince of Dragonstone, a power in his own right.

Also, the War did not start immediately after the kidnapping. There was this whole period where Brandon went to KL, King summoned Lord Rickard, Lord Rickard came to KL from Winterfell, Both were executed. King then sent orders to Lord Arryn to kill his wards. That started the war. So there must have been months in the prelude in which it wasnt clear if there was going to be a war so why should the cautious Prince Doran take the risk of offending the crown prince?

Prince Doran was however not pleased with the news he had received about the Stark girl and Rhaegar. When Aerys called his banners against the Rebels, Prince Doran was deliberately slow in response. He did not, if he knew Rhaegar was in Dorne, give command of Dornish Armies to the Prince. He only sent the Dornish Armies forward once the Crown Prince had left Dorne. There was this whole embassy under the Prince's uncle and King's guard Lewyn Martell too so it is plausible that it was Prince Lewyn who persuaded Doran to send his Armies forward.

In any case:

1. There is no indication that Prince Doran knew about the prophecy of The Prince that was promised.
2. There is no indication that Prince Doran was aware of Rhaegar's presence in the Red Mountains. Even the King didn't know about his son's whereabouts.
3. Prince Doran obviously felt insult by events of Tourney of Harrenhal which resulted in passive-aggressive act of defiance e.g. not giving Rhaegar the command of his armies and being slow in his reaction when commanded by the King to send forth the Armies of Dorne.

Answer 2

Lyanna and Rhaegar eloped to Dorne to be wed in secret.

In the Season 7 finale of Game of Thrones, it is revealed that Rhaegar and Lyanna loved each other and eloped. The ceremony, as described by Gilly earlier in the season, was in Dorne. This was presumably due to the enmity between the houses making it a better idea to leave the area -- in fact, Lyanna's father Rickard reacted to word of his daughter's being taken by Rhaegar very poorly; he stormed to King's Landing and demanded her return from Aerys, leading to his burning and finally igniting Robert's Rebellion.

Answer 3

I personally think the Elia was in on it. She knew and was Ok with it. I think Elia knew are encouraged Rhaegar to elope with Lyanna Stark. We know that Elia knew of the prophecy of the "Three-Heads" of the Dragon. Also Elia knew she couldn't give Rhaegar any more children, so maybe she suggested Rhaegar conceive the "third" head with another women. And preferably marry her, since Targaryens had practice polygamy. That's why I think Rhaegar took Lyanna to Dorne because Elia knew they would be safe in Dorne. I also believe it was Elia who told Rhaegar about the Tower of Joy. Of course, this is pure speculation of my part, but It would make sense (At least to me).