  A numerical method is developed to efficiently calculate the stress (and
displacement) field in finite 2D rectangular media. The solution is expanded on
a function basis with elements that satisfy the Navier-Cauchy equation. The
obtained solution approximates the boundary conditions with their finite
Fourier series. The method is capable to handle Dirichlet, Neumann and mixed
boundary value problems as well and it was found to converge exponentially fast
to the analytical solution with respect to the size of the basis. Possible
application in discrete dislocation dynamics simulations is discussed and
compared to the widely used finite element methods: it was found that the new
method is superior in terms of computational complexity.
