Productions of Ibsen always work best when they demonstrate just how relevant and universal the themes within his plays are. And in Elinor Cook's strong new adaptation, directed here by soon-to-be Young Vic boss Kwame Kwei-Armah, the fundamental thing you take away from The Lady from the Sea is how it could have been written yesterday. Not bad for a 19th century Norwegian playwright.

The Lady from the Sea paints an astonishingly astute and dynamic portrayal of relationships, looking hardest at the effect of a male-controlled society, where the benefits of marriage are reserved mainly for men. Yes, some may say we've come a long way since a wife was something to be owned. But the play's through-line that marriage should be about equality – freedom of decision, working together and truth – is absolutely still something for today.

Cook transports the action from the icy fjords in 1888 to a Caribbean island in the 1950s, which adds an apt claustrophobic heat, emphasised by the heavy scent of hibiscus hanging over Tom Scutt's boat-like set, but not much else. The Lady from the Sea follows Ellida Wangel, married to Doctor Wangel, on a day his two daughters from his previous marriage are celebrating the memory of their dead mother. Ellida, it transpires, has been out of sorts for some time and eventually she tells her husband that when she was a teenager, she was in love with a sailor. This sailor promised he would return for her, and 20 years later – now she has finally married – she believes he is coming back for her.

In a way, it is a simple, domestic plot, and the beauty of Cook's adaptation is that it keeps to this simplicity but draws out key themes such as the initially stark but ultimately balming effect of truth and the raw almost physical need for personal freedom. These come through the plot of Ellida and the doctor but also in the subplot of his daughter Bolette, who longs to go to university but is restricted by the guilt she feels at the idea of leaving her father. When her old tutor comes to stay, she is encouraged to start making her own life, but not before she is asked to relinquish something of herself first.

The show takes a little while to warm up, and in early scenes the acting is a little uneven, but Nikki Amuka-Bird as Ellida is magnetic, embodying a gentle, poised longing and deep-seated unhappiness. She smiles a lot, but it is a smile that hides a storm of torment. Her scenes with Finbar Lynch are very well pitched, and Lynch beautifully demonstrates the doctor's incomprehension that his increasing remoteness from his wife may not be something he can 'cure', like a broken leg.

Scutt's set places a body of water, Ellida's beloved sea, in one corner of the stage and it is the sound and texture of this which adds richness to the piece. Kwei Armah's steady directing hand focuses on the scenes between Lynch and Amuka-Bird, who allows Cook's adaptation to fly. It is an unshowy, almost discrete, production, which lets Ibsen's astonishing vision speak with clarity. Kwei-Armah gives the play space to be what it is: a beautiful, delicate and universal portrayal of human relationships at their most complex.

The Lady from the Sea runs at the Donmar Warehouse until 2 December.