Matrix solvers for flow simulation by the finite element method

Following the finite element discretization of the Navier-Stokes equations we obtain a nonlinear matrix system. The resolution of this system by the iterative method of Newton-Raphson requires at each iteration the factorization of a matrix of the same size and the same structure as the matrix of the system. The storage space required for storage and the machine time required for their factorization become excessive. For industrial problems (3D problems or even in 2D), It is practically impossible to work with matrices, the memory space to store them is out of reach of current computers. To overcome this problem, the linear systems resulting from the Newton-Raphson algorithm are solved by the GMRES (Generalized Minimal RESidual) algorithm. The structure of the GMRES algorithm makes that we never have matrices to evaluate, but only the action of a matrix on a vector. Convergence is accelerated using a pre-conditioning technique. The diagonal pre-conditioning that we have developed is simple and effective.