SITTING in a private member's club in Cannes, Viggo Mortensen is struggling, stifling yawns and drawing on his packet’s last cigarette. He can be forgiven for flagging. The night before, the 57 year-old’s latest film Captain Fantastic premiered at the film festival to an overwhelming response. "You could see it, you could feel it," he says, in soft tones. "People really loved the movie and they were drawn to it. They wanted to talk about it." It meant a long night of champagne and celebration.

This isn't his first rodeo, as they say. The Danish-American Mortensen has been to Cannes on a number of occasions. He and his fellow Middle Earth-dwellers came to the festival to bring twenty-minutes of sneak footage of Peter Jackson's Lord of the Rings – the Tolkien trilogy that forever cemented him in the public consciousness as the warrior Aragorn. Later, he returned for A History of Violence, the first of three fabulous collaborations in six years with David Cronenberg (Eastern Promises and A Dangerous Method followed).

While subsequent Cannes appearances included On The Road – the response was "lukewarm" to the Jack Kerouac adaptation, he concedes – he can't be too surprised at the Captain Fantastic reception. Since debuting at Sundance in January, it's been building momentum, capturing hearts wherever it's played. In Cannes, a few days after we meet, it won writer-director Matt Ross the Un Certain Regard directing award. Now it's approaching the awards season as one of the year's must-see indies – like Room, Whiplash or Beasts of the Southern Wild before it.

For Mortensen, who may well add to his one Oscar nomination (for his Russian gangster in Eastern Promises), it's yet another meticulously crafted performance. Here, he plays Ben, an unconventional father of six who is raising his children in the wilds of America's Pacific Northwest. He not only teaches them survival skills – including one remarkable rock climbing sequence – but encourages them to read philosophers and thinkers like Noam Chomsky. But is it the right way to raise his kids and let them prepare for life?

"I think he’s an involved, absolutely committed, loving father," says Mortensen. "Does he go too far, does he make mistakes? Yes. That makes him even more human and interesting. And what ultimately makes him a really good father is the ability to finally recognise when he’s wrong and to make an effort to make amends for it. Or change course, adapt, be flexible. He encourages dialogue, he encourages people to think for themselves, and in the end, he makes an effort to strike a new balance within himself and his family."

Mortensen has a 28 year-old son, Henry, from his past marriage to actress/singer Exene Cervenka, so was Ben's way of parenting something he related too? “I have always had a discourse with my son in that sense. We had very adult conversations when he was little – also because he had an aptitude for it and an interest in it. We talked about very serious subjects very early on. I think Matt [Ross] has a similar relationship with his son. But [it’s really about] caring enough and making time, even when there doesn’t seem to be time."

He then proceeds to given an anecdote which not only illuminates the father-son dynamic in the Mortensen household but the actor's approach to his work. "I’d be driving him to school, saying some sentence over and over again with some strange accent. Then I might just jokingly say to him – with a Russian accent – ‘I’ll be picking you up after school!’ I was just playing with it, but he’s had more than a glimpse into what it is to prepare and work on a movie."

Mortensen is well-known for his prep work. On Lord of the Rings, he spent time living in the woods to play Aragorn in his costume, never taking it off (he stopped short of sleeping with his sword, as some have said). For the post-apocalyptic Cormac McCarthy adaptation The Road, he soaked his shoes in water before going on camera and slept outsider under a tarpaulin. And to play Sigmund Freud in A Dangerous Method, he haunted bookstores, collecting texts the famed psychoanalyst would've had in his library, and visited Vienna, "walking where he walked".

Similarly, for Captain Fantastic, he took to the forests – never once sleeping in the hotel room the production had hired for him. "I like the woods," he shrugs, noting that his father, who grew up in the Danish countryside, taught the young Viggo to hunt, fish and camp. While he was born in New York – his mother's American – he spent his childhood all over, from Denmark to Venezuela and Argentina, before his parents divorced when he was 11. Even now, he's a citizen of the world, living in Spain with his actress-girlfriend Ariadna Gil.

With all this prep, it's clear why Mortensen doesn't make more than one movie a year. He has other interests to pursue – painting, poetry and running his boutique publishing firm Perceval Press. But it’s more than that. "When you first start out as an actor, you just want to do any kind of work – to try and make a living," he says, recalling those early days when he pitched up in Peter Weir's 1985 Amish thriller Witness. "But for many years now, I’ve been able to wait to find the right thing. I’m not looking for what could pay me the most or what could win me an award. If only all actors had this much integrity.

Captain Fantastic opens on September 9