2D Linear elasticity. Weighted Jacobi relaxation
For small deformations the equilibrium equation in the displacement formulation is 
(μ + λ)∇divu + μΔu
+ f = 0,
where u = (X, Y) is displacement of a point from equilibrium position,
μ, λ are Lame parameters (we put μ = 1 ).
Using 2-nd order finite differences we obtain the approximate equation
(2 + λ)(Xi+1,j + Xi-1,j - 2Xi,j ) +
Xi,j+1 + Xi,j-1 - 2Xi,j + (Yi+1,j+1 + Yi-1,j-1
- Yi+1,j-1 - Yi-1,j+1 )(1 + λ)/4 + h2Fxi,j = 0.
2D weighted Jacobi relaxation algorithm
For weighted Jacobi relaxation of (Xi,j ,
Yi,j ) on n×n grid we rewrite equations (*) as
2(3 + λ)Xi,j = (2 + λ)(Xi+1,j + Xi-1,j ) +
Xi,j+1 + Xi,j-1 + (Yi+1,j+1 + Yi-1,j-1
- Yi+1,j-1 - Yi-1,j+1 )(1 + λ)/4 + h2Fxi,j
See also 2D weighted Jacobi relaxation on CPU.
 Linear elasticity (Wiki)
Simulations on GPU
updated 7 Aug 2012