Preconditioning by approximations of the Gram matrix for convection–diffusion equations

The paper analyses the numerical performance of preconditioning with Gram matrix approximations for the solution of a convection–diffusion equation. The convection–diffusion equation is discretized on a rectangular grid by standard finite element methods with piecewise linear test and trial functions. The discrete linear system is solved by two different conjugate gradient algorithms: CGS and GMRES. The preconditioning with Gram matrix approximations consists of replacing the solving of the equation with the preconditioner by a few iterations of an appropriate iterative scheme. Two iterative algorithms are tested: incomplete Cholesky and multigrid. Numerical experiments indicate that these preconditioners are efficient at relatively small values of the Reynolds number.