An element-by-element preconditioned conjugate gradient method implemented on a vector computer

We consider the linear equation Ax = b where A is a sparse symmetric positive definite matrix arising from a finite element discretisation. We use the preconditioned conjugate gradient method to solve this equation, introducing an element-by-element preconditioner which is based on a Crout's decomposition of the element matrices and an element-by-element product of them. When the mesh is coloured, this preconditioner is largely vectorizable. We implement this method on a CRAY-2, and test it on 2D and 3D elastic and thermal problems and compare it to other classical preconditioners.