Coming from CFD, I'm trying to understand solid mechanics. Currently programmed a simple linear elasticity model with equation

div(tau) + f = 0.

The problem is solved with simplest finite differences on a square mesh since the aim is to make an efficient GPU implementation.
I'm using displacement formulation. When I have boundary conditions for displacement, it's ok. When I have them for stress, *how do I find boundary displacement then*? I can get strain from known stress, but how to get displacement then, some kind of numerical integration?