Randomized vector Gauss-Seidel algorithm with black-red ordering for solving the elastostatics Lamé equation

A stochastic iterative algorithm for solving the elastostatics Lamé equation in a two-dimensional domain is suggested. The Dirichlet boundary value problem for a system of two coupled second order elliptic equations for the displacement vector components is considered. We approximate the Lamé equations with finite differences and transform the resulting system of linear algebraic equations using the red-black ordering of the Gauss-Seidel method. We solve this system using a vector randomized algorithm. The idea behind these stochastic methods is a randomized vector representation of matrix iterations that are performed by sampling random columns only, avoiding matrix by matrix and matrix by vector multiplications. We test the developed iterative algorithm by a comparison of the simulation results with the exact solution of the Lamé equation.